RD Sharma Class 12 Exercise 8.2 Continuity Solutions Maths - Download PDF Free Online

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RD Sharma Class 12 Exercise 8.2 Continuity Solutions Maths - Download PDF Free Online

RD Sharma Class 12 Exercise 8.2 Continuity Solutions Maths - Download PDF Free Online

Kuldeep MauryaUpdated on 27 Jan 2022, 02:13 PM IST

RD Sharma is one of the most well-known books in the country. RD Sharma's books are detailed, informative, and also contain step-by-step solutions for their problems. Class 12 RD Sharma chapter 8 exercise 8.2 solution deals with the chapter 'Continuity.' The book has plenty of examples that the students can practice to develop their skills. Still, when it comes to solving a complex chapter like Continuity, students also need to exercise problems. This is where RD Sharma class 12th exercise 8.2 solution comes to light.

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RD Sharma Class 12 Solutions Chapter 8 Continuity- Other Exercise
Continuity Excercise:8.2
RD Sharma Chapter wise Solutions

Also Read - RD Sharma Solution for Class 9 to 12 Maths

RD Sharma Class 12 Solutions Chapter 8 Continuity- Other Exercise

Continuity Excercise:8.2

Continuity exercise 8.2 question 1

Answer:
f(x) is everywhere continuous.
Hint:
A function is everywhere continuous when it is continuous at every x$\in$IR
Given:
$f(x)=\left\{\begin{array}{cc} \frac{\sin x}{x} & x<0 \\ x+1 & x \geq 0 \end{array}\right.$
Explanation:
Now, at x = 0
$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} \frac{\sin x}{x} \quad\left[\because \lim _{h \rightarrow 0} \frac{\sin x}{x}=1\right] \\ &=1 \\ &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} x+1=1 \end{aligned}$
And
$f(x) = 0 + 1 = 1$
As,
$\left.\lim _{x \rightarrow 0} f(x)=f(0) \quad \text { [for continuity } \lim _{x \rightarrow 0} f(x)=f(0)\right]$
Hence, f(x) is everywhere continuous.
Note: sine function, identity function, polynomial functions are everywhere continuous.

Continuity exercise 8.2 question 2

Answer:
Discontinuous at x = 0.
Hint:
If a function is not continuous at one point then it is discontinuous. As at that point
$\lim _{x \rightarrow a^{-}} f(x) \neq \lim _{x \rightarrow a^{+}} f(x) \neq f(0)$
and this is the definition of discontinuity.
Given:
$f(x)= \begin{cases}\frac{x}{|x|} & x \neq 0 \\ 0 & x=0\end{cases}$
Explanation:
Now consider at x = 0
$\begin{aligned} &\text { L.H.L } \lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{-h}{-h \mid}=-1 \\ &\text { R.H.L } \lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{h}{|h|}=1 \end{aligned}$
So
$L.H.S \neq R.H.L$
Function is discontinuous at X = 0

Continuity exercise 8.2 question 3 (i)

Answer:
x = 1
Hint:
Show LHS $\neq$ RHS or LHL $\neq$ the value of function at given point or RHL $\neq$ the value of function at given point.
Given:
$f(x)=\left\{\begin{array}{rr} x^{3}-x^{2}+2 x-2 & x \neq 1 \\ 4 & x=1 \end{array}\right.$
Explanation:
Now consider the point x = 1
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 1} f(x)=\lim _{x \rightarrow 1} x^{3}-x^{2}+2 x-2 \\ &=1-1+2 \times 1-2 \\ &=0 \end{aligned}$
$\begin{aligned} &\text { R.}{\text {H.L }}=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0}(1+h)^{3}-(1-h)^{2}+2(1+h)-2 \\ &=1-1+2-2=0 \\ &\qquad f(1)=4 \end{aligned}$
As
$\begin{aligned} &\text { L.H.L } = \text { R.H.L }\neq f(1) \end{aligned}$
f(x) is discontinuous at x =1

Continuity exercise 8.2 question 3 (ii)

Answer:
x = 2
Hint:
To check the continuity of such type of function we check at the breaking point.
Given:
$f(x)= \begin{cases}\frac{x^{4}-16}{x-2} & x \neq 2 \\ 16 & x=2\end{cases}$
Explanation:
At point x =2
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0}(2-h)=\lim _{h \rightarrow 0} \frac{(2-h)^{4}-16}{(2-h)-2} \\ &=\lim _{h \rightarrow 0} \frac{2^{4}-4.8 h+6.4 h^{2}-4.2 h^{3}+h^{4}-16}{-h} \end{aligned}$
$\begin{aligned} &=\lim _{h \rightarrow 0} \frac{32-32 h+24 h^{2}-8 h^{3}+h^{4}-16}{-h} \\ &=\lim _{h \rightarrow 0} 32-24 h+8 h^{2}-h^{3}=32 \end{aligned}$
$\text { R.H.L }=\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0}(2+h)=\lim _{h \rightarrow 0} \frac{(2+h)^{4}-16}{(2+h)-2}$
$\begin{aligned} &=\lim _{h \rightarrow 0} \frac{2^{4}+4.8 h+6.4 h^{2}+4.2 h^{3}+h^{4}-16}{h}=32 \\ &\text { Also } f(2)=16 \\ &\text { Thus } L . H .L=R . H . L \neq f(2) \end{aligned}$
Therefore, f(x) is discontinuous at x = 2

Continuity exercise 8.2 question 3 (iii)

Answer:
x = 0
Hint:
$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
Given:
$f(x)= \begin{cases}\frac{\sin x}{x} & x<0 \\ 2 x+3 & x \geq 0\end{cases}$
Explanation:
At x = 0
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{\sin (-h)}{-h} \\ &=\lim _{h \rightarrow 0} \frac{-\sinh }{-h}=1 \end{aligned}$
$\begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{\sinh }{h}=1 \\ &\text { Also } f(0)=2 \times 0+3=3 \\ &\text { Thus L.H.L}=\text {R.H.L} \neq \mathrm{f}(0) \end{aligned}$
f is discontinuous at x = 0

Continuity exercise 8.2 question 3 (iv)

Answer:
x = 0
Hint:
$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
Given:
$f(x)= \begin{cases}\frac{\sin 3 x}{x} & x \neq 0 \\ 4 & x=0\end{cases}$
Explanation:
At x = 0
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{\sin 3(-h)}{-h} \\ &=\lim _{h \rightarrow 0} \frac{-\sin 3 \mathrm{~h}}{-h}=1 \\ &=\lim _{h \rightarrow 0} \frac{-3 \sin 3 \mathrm{~h}}{3 h}=3 \end{aligned}$
$\begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{\sin 3 \mathrm{~h}}{h}=3 \\ &\text { Also } f(0)=4 \\ &\text { Thus L.H.L }=\text { R.H.L } \neq \mathrm{f}(0) \end{aligned}$
$\therefore$ f is not continuous at x = 0.

Continuity exercise 8.2 question 3 (v)

Answer:
x = 0
Hint:
$\lim _{x \rightarrow a} f(x)+g(x)=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x)$
Given:
$f(x)= \begin{cases}\frac{\sin x}{x}+\cos x & x \neq 0 \\ 5 & x=0\end{cases}$
Explanation:
At x = 0
$\begin{aligned} &\text { L.H. } L=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{\sin (-h)+\cos (-h)}{-h} \\ &=\lim _{h \rightarrow 0} \frac{-\sinh }{-h}+\cosh =1+1=2 \end{aligned}$
$\begin{aligned} &R . H \cdot L=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{\sinh }{h}+\cosh =1+1=2 \\ &\text { Also } f(0)=5 \\ &\text { Thus } L . H . L=R . H . L \neq \mathrm{f}(0) \end{aligned}$
As
$\lim _{x \rightarrow 0} f \rightarrow(x) \neq x f(0)$
f is not continuous at x = 0

Continuity exercise 8.2 question 3 (vi)

Answer:
x = 0
Hint:
Use L-Hospital Rule when we find 0/0 form
Given:
$f(x)= \begin{cases}\frac{x^{4}+x^{3}+2 x^{2}}{\tan ^{-1} x} & x \neq 0 \\ 10 & x=0\end{cases}$
Explanation:
at x = 0
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{(-h)^{4}+(-h)^{3}+2(-h)^{2}}{\tan ^{-1}(-h)} \\ &=\lim _{h \rightarrow 0} \frac{h^{4}-h^{3}+2 h^{2}}{\tan ^{-1}(h)}=0 \end{aligned}$
$\begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{h^{4}+h^{3}+2 h^{2}}{\tan ^{-1}(h)}=0 \\ &\text { Also } f(0)=10 \\ &\text { Thus L.H.L } =\text { R.H.L } \neq f(0) \end{aligned}$
Hence, f(x) is discontinuous at x = 0.

Continuity exercise 8.2 question 3 (vii)

Answer:
x = 0
Hint:
Use L-Hospital Rule when we find 0/0 form
Given:
$f(x)=\left\{\begin{array}{cl} \frac{e^{x}-1}{\log _{e}(1-2 x)} & x \neq 0 \\ 7 & x=0 \end{array}\right.$
Explanation:
At x = 0
$\text { L.H.L } =\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{e^{-h}-1}{\log _{e}(1-2 h)}$
$=\lim _{h \rightarrow 0} \frac{\frac{e^{-h}-1}{-h}}{\frac{\log _{e}(1-2 h)}{-2 h} \times 2}=\frac{1}{2}$
$\text { R.H.L }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{e^{h}-1}{\log _{e}(1+2 h)}$
$=\lim _{h \rightarrow 0} \frac{\frac{e^{h}-1}{h}}{\frac{\log _{e}(2 h)}{2 h} \times 2}=\frac{1}{2}$
$\begin{aligned} &\text { Also } f(0)=7 \\ &\text { Thus L.H.L } =\text { R.H.L } \neq f(0) \end{aligned}$
$\therefore$f(x) is not continuous at x = 0

Continuity exercise 8.2 question 3 (viii)

Answer:
f(x) is nowhere discontinuous
Hint:
$f(x)=|x|= \begin{cases}x & x>0 \\ -x & x<0\end{cases}$
Given:
$f(x)=|x|= \begin{cases}|x-3| & x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{15}{4} & x<1\end{cases}$
Explanation:
$\begin{aligned} &f(x)= \begin{cases}x-3 & x \geq 3 \\ -(x-3) & 1 \leq x<3 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4} & x<1\end{cases} \end{aligned}$
Now, we check the limit at x = 1 and x = 3
At x = 3
$\begin{aligned} &\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3}-(x-3)=0 \\ &\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3} x-3=3-3=0 \\ &f(3)=3-3=0 \end{aligned}$
As
$\lim _{x \rightarrow 3} f(x)=f(3)$
Hence, f(x) is continuous at x = 3
Now, at x = 1
$\begin{aligned} &\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1} \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4} \\ &=\frac{1}{4}-\frac{3}{2}+\frac{13}{4} \\ &=2 \\ &\lim _{x \rightarrow 1^{+}} f(x)=-(1-3)=2 \\ &f(1)=-(1-3)=2 \end{aligned}$
As
$\lim _{x \rightarrow 1^{+}} f(x)=f(1)$
$\therefore$f(x) is continuous at x = 1
As polynomial function is everywhere continuous & greater integer function is continuous except its end point &
$\lim _{x \rightarrow 1} f(x) \text { \& } \lim _{x \rightarrow 3} f(x)$
is continuous.
Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 3 (ix)

Answer:
at x = 3
Hint:
Check the continuity of function at breaking points.
Given:
$f(x)=\left\{\begin{array}{cc} |x|+3 & x \leq-3 \\ -2 x & -3<x<3 \\ 6 x+2 & x>3 \end{array}\right.$
Explanation:
at x = -3
$\begin{aligned} &\lim _{x \rightarrow-3^{+}} f(x)=\lim _{x \rightarrow-3}-2 x=6 \\ &f(-3)=0+3+3=6 \\ &\lim _{x \rightarrow-3^{-}} f(x)=\lim _{x \rightarrow-3}-x+3 \\ &=6 \end{aligned}$
Now at x = 3
$\begin{aligned} &\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3}-2 x=-6 \\ &\quad \lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3} 6 x+2=20 \\ \end{aligned}$
As
$\begin{aligned} &\lim _{x \rightarrow 3^{-}} f(x) \neq \lim _{x \rightarrow 3^{+}} f(x) \end{aligned}$
f(x) is not continuous at x = 3.

Continuity exercise 8.2 question 3 (x)

Answer:
x = 1
Hint:
Check the continuity of function at end points.
Given:
$f(x)= \begin{cases}x^{10}-1 & x \leq 1 \\ x^{2} & x>1\end{cases}$
Explanation:
At x = 1
$\begin{aligned} &\lim _{x \rightarrow 1^{-}} x^{10}-1=\lim _{x \rightarrow 1} x^{10}-1 \\ &=(1)^{10}-1=1-1=0 \end{aligned}$
$\begin{aligned} &\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1} x^{2} \\ &=(1)^{2}=1 \\ \end{aligned}$
As
$\begin{aligned} &\lim _{x \rightarrow 1^{-}} f(x) \neq \lim _{x \rightarrow 1^{+}} f(x) \end{aligned}$
f(x) is not continuous at x = 1

Continuity exercise 8.2 question 3 (xi)

Answer:
x = 1
Hint:
Check at end points.
Given:
$f(x)=\left\{\begin{array}{cc} 2 x & x<0 \\ 0 & 0 \leq x \leq 1 \\ 4 x & x>1 \end{array}\right.$
Explanation:
At x = 0
$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} 2 x=2 \times 0=0-(1) \\ &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} 0=0-(2) \\ &f(0)=0-(3) \end{aligned}$
As (1)=(2)=(3)
Hence, f(x) is continuous at x = 0
At x = 1
$\begin{aligned} &\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1} 4 x=4 \times 1=4 \\ &f(1)=0 \end{aligned}$
As
$\lim _{x \rightarrow 1^{+}} f(x) \neq f(1)$
Hence, f(X) is discontinuous at x = 1

Continuity exercise 8.2 question 3 (xii)

Answer:
f(x) is continuous everywhere.
Hint:
$\lim _{x \rightarrow 0} f(x)-g(x)=\lim _{x \rightarrow 0} f(x)-\lim _{x \rightarrow 0} g(x)$
Given:
$f(x)=\left\{\begin{array}{cc} \sin x-\cos x & x \neq 0 \\ -1 & x=0 \end{array}\right.$
Explanation:
At x = 0
$\begin{aligned} &\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \sin x-\cos x \\ &=\lim _{x \rightarrow 0} \sin x-\lim _{x \rightarrow 0} \cos x \\ &=\sin 0-\cos 0 \\ &=-1 \end{aligned}$
And
$\begin{aligned} &f(x)=-1 \end{aligned}$
Constant function,
Sine and cosine function is everywhere continuous and f(x) is continuous at x = 0
$\therefore$f(x) is everywhere continuous.

Continuity exercise 8.2 question 3 (xiii)

Answer:
everywhere continuous.
Hint:
Constant and identity function is everywhere continuous.
Given:
$f(x)=\left\{\begin{array}{rr} -2 & x \leq-1 \\ 2 x & -1<x<1 \\ 2 & x \geq 1 \end{array}\right.$
Explanation:
Constant and identity function are everywhere continuous. So, we only have to check at x = -1, x = 1 (end points) -(1)
At x = -1
$\begin{aligned} &\lim _{x \rightarrow-1} f(x)=\lim _{x \rightarrow-1}-2=-2 \\ &f(-1)=-2 \\ &\lim _{\left(x \rightarrow 1^{+}\right)} f(x)=\lim _{x \rightarrow 1^{-}} 2 \times x \\ &=2 \times-1=-2 \end{aligned}$
As
$\begin{gathered} \lim _{x \rightarrow-1} f(x)=f(-1) \\ f(x)_{\text {is continuous at }} x=-1-(2) \end{gathered}$
At x = 1
$\begin{aligned} &\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1} 2 x=2 \times 1=2 \\ &\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1} 2=2 \\ &f(1)=2 \end{aligned}$
As
$\lim _{x \rightarrow 1} f(x)=f(1)-(3)$
So from (1), (2) & (3)
f(x) is everywhere continuous.

Continuity exercise 8.2 question 4 (i)

Answer:
$\frac{2}{15}$
Hint:
f(x) is continuous so it is also continuous at end points. Put LHL= RHL
Given:
$f(x)= \begin{cases}\frac{\sin 2 x}{5 x} & x \neq 0 \\ 3 k & x=0\end{cases}$
Explanation:
At x = 0
$\begin{aligned} &\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{\sin 2 x}{5 x} \\ &=\frac{1}{5} \lim _{x \rightarrow 0} \frac{\sin 2 x}{2 x} \times 2 \\ &=\frac{2}{5} \lim _{x \rightarrow 0} \frac{\sin 2 x}{2 x}\left[\operatorname{as} \lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right] \\ &=\frac{2}{5} \times 1=\frac{2}{5} \\ &\qquad f(0)=3 k \end{aligned}$
As f(X) is continuous at x = 0 when
$\begin{aligned} &\lim _{x \rightarrow 0} f(x)=f(0) \\ &\frac{2}{5}=3 k \\ &k=2 / 15 \end{aligned}$

Continuity exercise 8.2 question 4 (ii)

Answer:
$k = -2$
Hint:
Put LHL = RHL = f(2)
Given:
$f(x)= \begin{cases}k x+5 & x \leq 0 \\ x-1 & x>0\end{cases}$
Explanation:
LHL = RHL = f(2) ....(1)
$\begin{aligned} &\text { L.H.L } =\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0}(2-h)=\lim _{h \rightarrow 0} k(2-h)+5 \\ &=2 k+5 \end{aligned}$
$R . H . L=\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0}(2+h)=\lim _{h \rightarrow 0} k(2+h)-1=1$
Using (1) we get
$\begin{aligned} &\therefore 2 k+5=1 \\ &2 k=1-5 \\ &=-4 \\ &k=-2 \end{aligned}$

Continuity exercise 8.2 question 4 (iii)

Answer:
No value of k can make +
Hint:
Put L.H.L = R.H.L at x = 0
Given:
$f(x)= \begin{cases}k\left(x^{2}+3 x\right) & x<0 \\ \cos 2 x & x \geq 0\end{cases}$
Explanation:
At x = 0
L.H.L = R.H.L = f(0) ....(1)
$\begin{aligned} &\text { L.H.L } =\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} k\left[(-h)^{2}+3(-h)\right] \\ &=\lim _{h \rightarrow 0} k\left[h^{2}-3 h\right]=0 \\ &f(0)=\cos 2 \times 0=\cos 0=1 \\ &\text { L.H.L } \neq \mathrm{f}(0) \end{aligned}$
Hence no value of k can make f continuous.

Continuity exercise 8.2 question 4 (iv)

Answer:
$a=7 / 2, \: \: b=-17 / 2$
Hint:
Put LHL = RHL at x = 3 , 5
Given:
$f(x)= \begin{cases}2 & x \leq 3 \\ a x+b & 3<x<5 \\ 9 & x \geq 5\end{cases}$
Explanation:
At x = 3
$\begin{aligned} &\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3} a x+b\\ &=3 a+b\\ &f(3)=2 \qquad ....(1) \end{aligned}$$\begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 3^{+}} f(x)=\lim _{h \rightarrow 0}(3+h)=\lim _{h \rightarrow 0} a(3+h)+b \\ &=3 a+b \end{aligned}$

Now, f(x) is continuous at x = 3

If 3a + b =2 .....(A)

Now, at x = 5

$\begin{aligned} &\lim _{x \rightarrow 5^{-}} f(x)=\lim _{x \rightarrow 5} a x+b \\ &=5 a+b \\ & f(5)=9 \end{aligned}$

$\begin{aligned} &\text { L.H.L } =\lim _{x \rightarrow 5} f(x)=\lim _{h \rightarrow 0}(5-h)=\lim _{h \rightarrow 0} a(5-h)+b \\ &=5 a+b \end{aligned}$
Now, f(x) is continuous at x = 5
If 5a + b = 9 .....(B)
on solving A and B we get,
$a=7 / 2, \: \: b=-17 / 2$

Continuity exercise 8.2 question 4 (v)

Answer:
a = 3, b = 1
Hint:
Put LHL = RHL at x = 3 , 5
Given:
$f(x)= \begin{cases}4 & x \leq-1 \\ a x^{2}+b & -1<x<0 \\ \cos x & x \geq 0\end{cases}$
Explanation:
At x = -1
$\begin{aligned} &f(-1)=4\\ &\text { R.H.L }=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{h \rightarrow 0}(1-h)=\lim _{h \rightarrow 0} a(-1+h)^{2}+b=a+b\\ &\therefore a+b=4 \qquad ....(A) \end{aligned}$
Now at x = 0
$\begin{aligned} &f(0)=\cos 0=1 \\ &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} a(-h)^{2}+b \\ &=b \end{aligned}$
$\begin{aligned} &\therefore f(0)=L . H . L \\ &\Rightarrow b=1 \\ &\text { From }(A) \\ &a=3 \end{aligned}$
Hence, a = 3, b = 1

Continuity exercise 8.2 question 4 (vi)

Answer:
$p=-1/2$
Hint:
Put LHL = RHL at x = 0
Given:
$f(x)= \begin{cases}\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x} & -1 \leq x<0 \\ \frac{2 x+1}{x-2} & 0 \leq x \leq 1\end{cases}$
Explanation:
At x = 0
L.H.L
$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} \frac{\sqrt{1+p x}-\sqrt{1-p x}}{x} \\ &=\lim _{x \rightarrow 0} \frac{p}{\sqrt{1+p x}}+\frac{p}{2 \sqrt{1+p x}} \quad\left[0 / 0_{\text {from }}\right] \\ &=\lim _{x \rightarrow 0} \frac{2 p}{\sqrt{1+p x}} \times 2 \end{aligned}$
$\begin{aligned} &=\frac{2 p}{2 \times \sqrt{1+p \times 0}}=\frac{2 p}{2}=p \\ &R . H \cdot L=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{2 h+1}{h-2}=\frac{-1}{2} \end{aligned}$
As f(x) is continuous at x = 0
Hence, p = -1/2

Continuity exercise 8.2 question 4 (vii)

Answer:
a = 2, b = 1
Hint:
Put LHL = RHL at x = 2 &x = 10
Given:
$f(x)= \begin{cases}5 & x \leq 2 \\ a x+b & 2<x<10 \\ 21 & x \geq 10\end{cases}$
Explanation:
At x = 2
$\begin{aligned} &f(2)=5 \\ &\lim _{x \rightarrow 2^{+}} f(x)=i m_{x \rightarrow 2} a x+b \\ &=2 a+b \end{aligned}$
As, f(x) is continuous at x = 2 when 5 + 2a +b - (1)
At x = 10
$\begin{gathered} \lim _{x \rightarrow 10^{-}} f(x)=\lim _{x \rightarrow 10} a x+b \\ =10 a+b \\ f(10)=21 \end{gathered}$
As, f(x) is continuous at x = 10
10a + b =21
From (1) & (2) we have
8a = 16
a = 2
Put in (1)
b = 1
Hence, a = 2 & b = 1

Continuity exercise 8.2 question 4 (viii)

Answer:
k = 6
Hint:
Put LHL = RHL =
$f(\frac{\pi }{2})$
Given:
$f(x)= \begin{cases}\frac{k \cos x}{\pi-2 x} & x<\pi / 2 \\ 3 & x=\pi / 2 \\ \frac{3 \tan ^{2} x}{2 x-\pi} & x \geq 10\end{cases}$
Explanation:
$At\: \: x = \frac{\pi }{2}$
$\begin{aligned} &\text { L.H.L }=f(x) \text { at } x=\frac{\pi}{2} \text { is } \\ &=\lim _{x \rightarrow \frac{\pi}{2}} f(x) \\ &=\lim _{h \rightarrow 0}\left(h-\frac{\pi}{2}\right) \end{aligned}$
$=\lim _{h \rightarrow 0} \frac{k \cos \left(h-\frac{\pi}{2}\right)}{\pi-2\left(h-\frac{\pi}{2}\right)}=\frac{k}{2}$
$\begin{aligned} &\text { Again } \mathrm{f}\left(\frac{\pi}{2}\right)=3 \\ &\text { L.H.L } =f\left(\frac{\pi}{3}\right) \\ &\therefore \frac{k}{2}=3 \\ &k=6 \end{aligned}$

Continuity exercise 8.2 question 5

Answer:
$a=-1, b=1 \text { or } a=1, b=1 \pm \sqrt{2}$
Hint:
p at LHL = RHL at
$x =1 ,\: \sqrt{2}$
Given:
$f(x)= \begin{cases}\frac{x^{2}}{9} & 0 \leq x<1 \\ 9 & 1 \leq x<\sqrt{2} \\ \frac{2 b^{2}-4 b}{x^{2}} & \sqrt{2} \leq x<\infty\end{cases}$
Explanation:
$\begin{aligned} &\text { At } x=1, L . H . L=R . H . L=f(1) \quad ...(A)\\ &f(1)=A \quad ...(1) \end{aligned}$
$L . H . L=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0}(1-h)=\lim _{h \rightarrow 0} \frac{(1-h)^{2}}{a}=\frac{1}{a}$
Using (A)
$\begin{aligned} &a=\frac{1}{a} \Rightarrow a^{2}=1 \Rightarrow a=\pm 1\\ &\text { At } x=\sqrt{2}, \quad \text { L.H.L }=\text { R.H.L }=f(\sqrt{2}) \quad ....(B) \end{aligned}$
$f(\sqrt{2})=\frac{2 b^{2}-4 b}{(\sqrt{2})^{2}}=\frac{2 b^{2}-4 b}{2}=b^{2}-2 b \quad ....(2)$
$\text { L.H.L }=\lim _{x \rightarrow \sqrt{2}^{2}} f(x)=\lim _{h \rightarrow 0}(\sqrt{2}-h)=\lim _{h \rightarrow 0} a=a$
So using (B) we get
$\begin{aligned} &b^{2}-2 b=\pm 1 \\ &\qquad b^{2}-2 b=1 \text { or } \quad b^{2}-2 b=-1 \\ &\qquad b^{2}-2 b-1=0 \quad \text { or } \quad b^{2}-2 b+1=0 \end{aligned}$
$\begin{aligned} &b=\frac{1 \pm \sqrt{2}}{1} \quad \text { Or } \quad(b-1)^{2}=0\\ &b=1 \pm \sqrt{2} \quad \text { Or } \quad b=1 \end{aligned}$
Hence,
$a=-1, b=1 \quad \text { or } \quad a=1, b=1 \pm \sqrt{2}$

Continuity exercise 8.2 question 6

Answer:
$a = \pi /6,\; b=-\pi /12$
Hint:
Put at LHL = RHL at
$x = \pi /4,\; x=\pi /2$
Given:
$f(x)= \begin{cases}x+a \sqrt{2} \sin x & 0 \leq x<\pi / 4 \\ 2 x \cot x+b & \pi / 4 \leq x<\pi / 2 \\ a \cos 2 x-b \sin x & \pi / 2 \leq x<\pi\end{cases}$
Explanation:
At x = $\pi$/4
$L.H.L = R.H.L =f(\frac{\pi }{4}) \qquad ....(A)$
Now,
$\begin{aligned} &f\left(\frac{\pi}{4}\right)=2 \cdot \frac{\pi}{4} \cdot \cot \left(\frac{\pi}{4}\right)+b\\ &=\frac{\pi}{2} \cdot 1+b=\frac{\pi}{2}+b \qquad ....(1) \end{aligned}$
L.H.L
$\begin{aligned} &\lim _{x \rightarrow \pi / 4^{-}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{\pi}{4}-h\right)=\lim _{h \rightarrow 0}\left(\frac{\pi}{4}-h\right)+a \sqrt{2} \sin \left(\frac{\pi}{4}-h\right) \\ &=\frac{\pi}{4}+a \sqrt{2} \cdot \frac{1}{\sqrt{2}}=\frac{\pi}{4}+a \end{aligned}$
Using (A)
$\begin{aligned} &\frac{1}{2}+b=\frac{\pi}{4}+a\\ &a-b=\frac{\pi}{4} \qquad ....(B) \end{aligned}$
$\begin{aligned} &\text { At } x=\frac{\pi}{2} \\ &\text { L.H.L }=\text { R.H.L }=f\left(\frac{\pi}{2}\right) \quad \ldots(C) \end{aligned}$
Now
$f\left(\frac{\pi}{2}\right)=a \cos 2 \cdot \frac{\pi}{2}-b \sin \frac{\pi}{2}=-a-b \qquad ....(2)$
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow \frac{\pi^{-}}{2}} f(x)=\lim _{h \rightarrow 0}\left(\frac{\pi}{2}-h\right) \\ &=\lim _{h \rightarrow 0} 2\left(\frac{\pi}{2}-h\right) \cot \left(\frac{\pi}{2}-h\right)+b=b \end{aligned}$
Using (C)
$-a-b=b \Rightarrow 2 b=-a \Rightarrow b=\frac{-a}{2}$
From (B)
$\begin{aligned} &a+\frac{a}{2}=\frac{\pi}{4} \\ &\frac{3}{2} a=\frac{\pi}{4} \\ &\Rightarrow a=\frac{\pi}{6} \end{aligned}$
$\begin{aligned} &b=\frac{-a}{2}=-\frac{\pi}{12} \\ \end{aligned}$
Hence
$\begin{aligned} &a=\pi / 6, b=-\pi / 12 \end{aligned}$

Continuity exercise 8.2 question 6

Answer:
$a = \pi /6,\; b=-\pi /12$
Hint:
Put at LHL = RHL at
$x = \pi /4,\; x=\pi /2$
Given:
$f(x)= \begin{cases}x+a \sqrt{2} \sin x & 0 \leq x<\pi / 4 \\ 2 x \cot x+b & \pi / 4 \leq x<\pi / 2 \\ a \cos 2 x-b \sin x & \pi / 2 \leq x<\pi\end{cases}$
Explanation:
At x = $\pi$/4
$L.H.L = R.H.L =f(\frac{\pi }{4}) \qquad ....(A)$
Now,
$\begin{aligned} &f\left(\frac{\pi}{4}\right)=2 \cdot \frac{\pi}{4} \cdot \cot \left(\frac{\pi}{4}\right)+b\\ &=\frac{\pi}{2} \cdot 1+b=\frac{\pi}{2}+b \qquad ....(1) \end{aligned}$
L.H.L
$\begin{aligned} &\lim _{x \rightarrow \pi / 4^{-}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{\pi}{4}-h\right)=\lim _{h \rightarrow 0}\left(\frac{\pi}{4}-h\right)+a \sqrt{2} \sin \left(\frac{\pi}{4}-h\right) \\ &=\frac{\pi}{4}+a \sqrt{2} \cdot \frac{1}{\sqrt{2}}=\frac{\pi}{4}+a \end{aligned}$
Using (A)
$\begin{aligned} &\frac{1}{2}+b=\frac{\pi}{4}+a\\ &a-b=\frac{\pi}{4} \qquad ....(B) \end{aligned}$
$\begin{aligned} &\text { At } x=\frac{\pi}{2} \\ &\text { L.H.L }=\text { R.H.L }=f\left(\frac{\pi}{2}\right) \quad \ldots(C) \end{aligned}$
Now
$f\left(\frac{\pi}{2}\right)=a \cos 2 \cdot \frac{\pi}{2}-b \sin \frac{\pi}{2}=-a-b \qquad ....(2)$
$\begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow \frac{\pi^{-}}{2}} f(x)=\lim _{h \rightarrow 0}\left(\frac{\pi}{2}-h\right) \\ &=\lim _{h \rightarrow 0} 2\left(\frac{\pi}{2}-h\right) \cot \left(\frac{\pi}{2}-h\right)+b=b \end{aligned}$
Using (C)
$-a-b=b \Rightarrow 2 b=-a \Rightarrow b=\frac{-a}{2}$
From (B)
$\begin{aligned} &a+\frac{a}{2}=\frac{\pi}{4} \\ &\frac{3}{2} a=\frac{\pi}{4} \\ &\Rightarrow a=\frac{\pi}{6} \end{aligned}$
$\begin{aligned} &b=\frac{-a}{2}=-\frac{\pi}{12} \\ \end{aligned}$
Hence,
$\begin{aligned} &a=\pi / 6, b=-\pi / 12 \end{aligned}$

Continuity exercise 8.2 question 7

Answer:
a = 3, b = -2
Hint:
Put at LHL = RHL at x = 2, x = 4
Given:
$f(x)= \begin{cases}x^{2}+a x+b & 0 \leq x<2 \\ 3 x+2 & 2 \leq x \leq 4 \\ 2 a x+5 b & 4<x \leq 8\end{cases}$
Explanation:
At x = 2
$\begin{aligned} &L . H . L=R . H . L=f(2) \qquad \qquad \qquad...(A)\\ &f(2)=3 \times 2+2=8 \qquad \qquad \qquad...(1)\\ &\text { L.H. } L=\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0}(2-h)\\ &=\lim _{h \rightarrow 0}(2-h)^{2}+a(2-h)+b\\ &=4+2 a+b \end{aligned}$
from (B)
$\begin{aligned} &4+2 a+b=8\\ &\Rightarrow 2 a+b=4 \qquad \ldots(B)\\ &\text { Now at } x=4\\ &L . H . L=R . H . L=f(4) \quad \ldots (C)\\ &f(4)=3 \times 4+2=8 \quad \ldots (2) \end{aligned}$
$\begin{aligned} &R . H . L=\lim _{x \rightarrow 4^{+}} f(x)=\lim _{h \rightarrow 0}(4+h) \\ &=\lim _{h \rightarrow 0} 2 a(4+h)+5 b=8 b+5 h \end{aligned}$
From (C)
$8 b+5 h=14 \quad \dots(D)$
Solving (B) & (D) we get
Hence, a = 3, b = -2

Continuity exercise 8.2 question 8

Answer:
$\frac{1}{2}$
Given:
$f(x)= \begin{cases}\frac{\tan \left(\frac{\pi}{4}-x\right)}{\cot 2 x} & x \neq \pi / 4 \\ k & x=\pi / 4\end{cases}$
Hint:
Apply L-Hospital Rule when you get 0/0 from.
Explanation:
At x = $\pi$/4
$\begin{aligned} &\text { L.H.L } =\text { R.H.L }=f\left(\frac{\pi}{4}\right) \quad \ldots(A) \\ &\text { L.H.L }=\lim _{x \rightarrow \frac{\pi^{-}}{4}} f(x)=\lim _{h \rightarrow 0}\left(\frac{\pi}{4}-h\right) \end{aligned}$
$=\lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{4}-\frac{\pi}{4}+h\right)}{\cot \left(\frac{\pi}{4}-h\right)}=\lim _{h \rightarrow 0} \frac{\tanh }{\tan 2 h}$
$=\lim _{h \rightarrow 0} \frac{\frac{\tanh }{h}}{\frac{\tan 2 h}{2 h} \times 2}=\frac{1}{2}$
Hence F (x) will be continious on
$[0, \frac{\pi }{2}]$
If
$f(\frac{\pi }{4})=\frac{1}{2}$

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Continuity exercise 8.2 question 9

Answer:
Everywhere continuous
Given:
$f(x)=\left\{\begin{array}{cc} 2 x-1 & x<2 \\ \frac{3 x}{2} & x \geq 2 \end{array}\right.$
Hint:
Polynomial & identity functions are everywhere continuous.
Explanation:
As polynomial & identity function are everywhere continuous.
So, we only have to check at end point i.e. x = 2
$f(2)=\frac{3 \times 2}{2}=3$
$\begin{aligned} &\text { L.H.L } =\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0}(2-h)=\lim _{h \rightarrow 0} 2(2-h)-1=3 \\ &\text { R.H.L }=\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0}(2+h)=\lim _{h \rightarrow 0} \frac{3(2+h)}{2}=3 \\ &\text { L.H.L }=\text { R.H.L }=f(2)=3 \end{aligned}$
f(x) is continuous at x = 2 and everywhere

Continuity exercise 8.2 question 10

Answer:
everywhere continuous.
Hint:
sine function is everywhere continuous.
Given:
$f(x)=sin(x)$
Explanation:
$f(x)= \begin{cases}\sin x & x \geq 0 \\ \sin (-x) & x<0\end{cases}$
$= \begin{cases}\sin x & x \geq 0 \\ -\sin x & x<0\end{cases}$
As sine function is everywhere continuous &
$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0}-\sin 0=0 \\ &\text { and } \\ &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} \sin 0=0 \\ &f(0)=\sin 0=0 \end{aligned}$
Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 11

Answer:
Everywhere continuous
Hint:
sine function, polynomial function and identity are everywhere continuous.
Given:
$f(x)=\left\{\begin{array}{cc} \frac{\sin x}{x} & x<0 \\ x+1 & x \geq 0 \end{array}\right.$
Explanation:
At x = 0
$\begin{aligned} &f(0)=0+1=1 \\ &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h) \\ &=\lim _{h \rightarrow 0} \frac{\sin (-h)}{-h}=\lim _{h \rightarrow 0} \frac{-\sinh }{-h}=1 \\ &\text { R.H.L } =\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} h+1=1 \\ &\text { L.H.L } =\text { R.H.L }=f(0)=1 \end{aligned}$
So, f(x) is everywhere continuous.

Continuity exercise 8.2 question 12

Answer:
Discontinuous at all integral points.
Hint:
Greatest integer function is discontinuous at integral points.
Given:
$g(x)=x-[x]$
Explanation:
$g(x)=x-[x]$
It is defined at all integral points
let n be an integer.
Then,
$\begin{aligned} &g(n)=n-[n]=0\\ &\text { The left hand limit of } \mathrm{f} \text { at } \mathrm{x}=\mathrm{n} \text { is }\\ &\lim _{x \rightarrow n^{-}} g(x)=\lim _{x \rightarrow n^{n}}(x-[x])=\lim _{x \rightarrow n^{-}}(x)-\lim _{x \rightarrow n^{-}}[x]\\ &=n-(n-1)=1 \end{aligned}$
$\begin{aligned} &\text { The right hand limit of } f \text { at } x=n \text { is }\\ &\lim _{x \rightarrow n^{+}} g(x)=\lim _{x \rightarrow n^{+}}(x-[x])=\lim _{x \rightarrow n^{+}}(x)-\lim _{x \rightarrow n^{+}}[x]\\ &=n-n=0 \end{aligned}$
$\begin{aligned} &\text { L.H. } L \neq R . H . L\\ &\therefore g \text { is not continious at } x=n \end{aligned}$
Hence g is discontinious at all integral points.

Continuity exercise 8.2 question 13 (i)

Answer:
everywhere continuous.
Hint:
sine and cosine function are everywhere continuous.
Given:
$f(x)=sin\: x+cox\: x$
Explanation:
As sine and cosine function are everywhere continuous and addition of two continuous functions is again continuous.
Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 13 (ii)

Answer:
everywhere continuous.
Hint:
sine and cosine function are everywhere continuous.
Given:
$f(x)=sin \: x-cos\: x$
Explanation:
As sine and cosine function are everywhere continuous and subtraction of two continuous functions is continuous.
Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 13 (iii)

Answer:
everywhere continuous.
Hint:
sine and cosine function are everywhere continuous.
Given:
$f(x)=sin\: x \times cos\: x$
Explanation:
As sine and cosine function are everywhere continuous and product of two continuous functions is continuous.
Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 14

Answer:
cos² x is continuous
Given:
f(x) = cos x²
Explanation:
f(x) = cos x²
Let a be any real number then,
L.H.L
$\begin{aligned} &\lim _{x \rightarrow a-} f(x)=\lim _{h \rightarrow 0} f(a-h) \\ &=\lim _{h \rightarrow 0} \cos (a-h)^{2} \\ &=\cos a^{2} \end{aligned}$
R.H.L
$\begin{aligned} &\lim _{x \rightarrow a^{-}} f(x)=\lim _{h \rightarrow 0} f(a+h) \\ &=\lim _{h \rightarrow 0} \cos (a+h)^{2} \\ &=\cos a^{2} \end{aligned}$
Also, L.H.L=R.H.L=f(a)
f(x) is continuous everywhere

Continuity exercise 8.2 question 15

Answer:
|cos x| is continuous.
Hint:
Continuous of two continuous functions is continuous.
Given:
f(x) = |cos x|
Explanation:
f(x) = |cos x|
Let a be any real number
L.H.L
$\begin{aligned} &\lim _{x \rightarrow a^{-}} f(x)=\lim _{h \rightarrow 0} f(a-h) \\ &=\lim _{h \rightarrow 0}|\cos (a-h)| \\ &=|\cos a| \end{aligned}$
R.H.L
$\begin{aligned} &\lim _{x \rightarrow a^{+}} f(x)=\lim _{h \rightarrow 0} f(a+h) \\ &=\lim _{h \rightarrow 0}|\cos (a+h)| \\ &=\cos a \end{aligned}$
L.H.L = R.H.L = f(a)
f(x) is continuous everywhere

Continuity exercise 8.2 question 16

Answer:
No point of discontinuity.
Given:
$f(x)=\left | x \right |-\left | x+1 \right |$
Explanation:
$f(x)=\left | x \right |-\left | x+1 \right |$
$\begin{gathered} =\left\{\begin{array}{c} -x+x+1, x<-1 \\ -x-(x+1),-1 \leq x<0 \\ x-(x+1), x \geq 0 \end{array}\right\} \\ =\left\{\begin{array}{c} 1, x<-1 \\ -2 x-1,-1 \leq x<0 \\ -1, x \geq 0 \end{array}\right\} \end{gathered}$
Continuity at x = -1
L.H.L
$\begin{aligned} &\lim _{x \rightarrow-1^{-}} f(x) \\ &=\lim _{x \rightarrow-1^{-}} 1=1 \end{aligned}$
R.H.L
$\begin{aligned} &\lim _{x \rightarrow-1^{+}} f(x) \\ &\lim _{x \rightarrow-1^{+}}(-2 x-1) \\ &=-2 \times-1-1=1 \end{aligned}$
And
$\begin{aligned} &f(-1)=-2 \times-1-1=1 \\ &\lim _{x \rightarrow-1^{-}} f(x)=\lim _{x \rightarrow-1^{-}} f(x)=f(-1) \end{aligned}$
F(x) is continuous at x=-1
Continuous at x=0
L.H.L
$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}(-2 x-1) \\ &=-2 \times 0-1=-1 \end{aligned}$
R.H.L
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}-1=-1$
And
$f(0)=-2\times 0-1=-1$
L.H.L=R.H.L=f(0)
F(x) is continuous at x=0, hence continuous everywhere

Continuity exercise 8.2 question 17

Answer:
f(x) is continuous.
Hint:
Check at x = 0
Given:
$f(x)=\left\{\begin{array}{cc} x^{2} \sin 1 / x & x \neq 0 \\ 0 & x=0 \end{array}\right.$
Explanation:
$f(x)=\left\{\begin{array}{cc} x^{2} \sin 1 / x & x \neq 0 \\ 0 & x=0 \end{array}\right.$
f is defined at all points of the real line
Let c be a real number
Case 1:
$\begin{aligned} &\text { If } c \neq 0 \text { then } \mathrm{f}(c)=c^{2} \sin \frac{1}{c}\\ &\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left(x^{2} \sin \frac{1}{x}\right)=\lim _{x \rightarrow c}\left(x^{2}\right) \cdot \lim _{x \rightarrow c}\left(\sin \frac{1}{x}\right)\\ \end{aligned}$
$\begin{aligned} &=c^{2} \sin \frac{1}{c}\\ &\therefore \lim _{x \rightarrow c} f(x)=f(c)\\ &\therefore f \text { is continious at all points } \mathrm{x} \neq 0 \end{aligned}$
Case 2:
$\begin{aligned} &\text { If } \mathrm{c}=0 \text { then } \mathrm{f}(0)=0\\ &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left(x^{2} \sin \frac{1}{x}\right)\\ &\text { It is known that }-1 \leq \sin \frac{1}{x} \leq 1, x \neq 0\\ &\Rightarrow-x^{2} \leq x^{2} \sin \frac{1}{x} \leq x^{2} \end{aligned}$
$\begin{aligned} &\Rightarrow-x^{2} \leq x^{2} \sin \frac{1}{x} \leq x^{2} \\ &\Rightarrow \lim _{x \rightarrow 0}\left(-x^{2}\right) \leq \lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right) \leq \lim _{x \rightarrow 0} x^{2} \end{aligned}$
$\begin{aligned} &\Rightarrow 0 \leq \lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right) \leq 0 \\ &\Rightarrow \lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right)=0 \\ &\therefore \lim _{x \rightarrow 0^{-}} f(x)=0 \end{aligned}$
Similarly,
$\begin{aligned} &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}\left(x^{2} \sin \frac{1}{x}\right) \\ &=\lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right)=0 \\ &\therefore \lim _{x \rightarrow 0^{-}} f(x)=f(0)=\lim _{x \rightarrow 0^{+}} f(x) \end{aligned}$
Therefore f is continuous at x=0
From the above observations.
It can be conclude that f is continuous at every point of the real line.
Thus f is a continuous function.

Continuity exercise 8.2 question 18

Answer:
$x=-2, -5/2$
Hint:
$f \circ f(x)=f(f(x))$
Given:
$f(x)=1 / x+2$
Explanation:
Clearly,
$f(x)=1 / x+2$
is discontinuous at
X = -2
Also, it is not defined at x = -2
For x ≠ -2
$\begin{aligned} &f(x)=f\left(\frac{1}{x+2}\right) \\ &=\frac{1}{\frac{1}{x+2}+2} \\ &=\frac{\frac{1}{2 x+5}}{x+2} \\ &=\frac{x+2}{2 x+5} \end{aligned}$
We observe that,
f(f(x)) is discontinuous and not defined a
$\begin{aligned} x=-\frac{5}{2} \end{aligned}$
Hence
f(f(x)) is not continuous at x = -2 and
$\begin{aligned} x=-\frac{5}{2} \end{aligned}$

Continuity exercise 8.2 question 19

Answer:
Discontinuous at x = 1/2 , 1 , 2
Hint:
f(x)/g(x) is continuous at every point hen f(x) & g(x) are continuous except
$g(x) \neq 0$
Given:
$f(x)=\frac{1}{t^{2}+t-2}, t=\frac{1}{x-1}$
Explanation:
$\begin{aligned} &f(t)=\frac{1}{t^{2}+t-2} \text { where } t=\frac{1}{x-1}\\ &\text { Clearly } t=\frac{1}{x-1} \text { is discontinuous at } \mathrm{x}=1 \end{aligned}$
$\begin{aligned} &\text { For } \mathrm{x} \neq 1 \text { we have, }\\ &f(t)=\frac{1}{t^{2}+t-2}=\frac{1}{(t+2)(t-1)} \end{aligned}$
$\begin{aligned} &\text { This is discontinuous at } x=-2 \text { and } \mathrm{t}=1\\ &\text { For } t=-2, t=\frac{1}{x-1} \Rightarrow x=\frac{1}{2}\\ &\text { For } t=1, t=\frac{1}{x-1} \Rightarrow x=2 \end{aligned}$
Hence F is discontinuous at
$\begin{aligned} x=\frac{1}{2}, x = 1 \text { and } x = 2 \end{aligned}$

The RD Sharma class 12 solution of Continuity exercise 8.2 consists of 40 questions that cover up almost the majority of all the topics of the chapter continuity. The concept covered in this chapter are-

Continuous function.
Absolute continuous function.
Absolute Continuity of a measure concerning another measure.
Continuous probability distribution.

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