Limits And Derivatives Class 11th Notes - Free NCERT Class 11 Maths Chapter 13 notes - Download PDF

Algebra of Limits $\lim\limits _{x \rightarrow a} f(x)$

Theorem 1:

If f and g are two functions where both limits $\lim\limits _{x \rightarrow a} f(x)$ and $\lim\limits _{x \rightarrow a} g(x)$ exist.

i) The limit of the sum of two functions is equal to the limit of the individual sums of each function.

$\lim\limits _{x \rightarrow a}[f(x)+g(x)] = \lim\limits _{x \rightarrow a} f(x) + \lim\limits _{x \rightarrow a} g(x)$

ii) The limit of the difference between two functions is equal to the limit of the individual differences of each function.

$\lim\limits _{x \rightarrow a}[f(x)-g(x)] = \lim\limits _{x \rightarrow a} f(x) - \lim\limits _{x \rightarrow a} g(x)$

iii) The limit of the product of two functions is equal to the limit of the individual product of each function.

$\lim\limits _{x \rightarrow a}[f(x).g(x)] = \lim\limits _{x \rightarrow a} f(x) . \lim\limits _{x \rightarrow a} g(x)$

iv) The limit of the quotient of two functions is equal to the quotient of the limit of each function.

$\lim\limits _{x \rightarrow a}[\frac{f(x)}{g(x)}] = \frac{\lim\limits _{x \rightarrow a} f(x)}{\lim\limits _{x \rightarrow a} g(x)}$ [Provided $\lim\limits _{x \rightarrow a} g(x)\neq0$]

v) $\lim\limits _{x \rightarrow a} (λ.f(x))=λ.\lim\limits _{x \rightarrow a} f(x)$

Limits of polynomials and rational functions

If $f$ is a polynomial function, then $\lim\limits _{x \rightarrow a} f(x)$ exists and is given by

$\lim\limits _{x \rightarrow a} f(x)=f(a)$

An important limit, which is very useful and used in the sequel, is given below:

$\lim\limits _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}$

Remark: The above expression remains valid for any rational number provided that ' $a$ ' is positive.
Limits of trigonometric functions
To evaluate the limits of trigonometric functions, we shall make use of the following limits, which are given below:

(i) $\lim\limits _{x \rightarrow 0} \frac{\sin x}{x}=1$
(ii) $\lim\limits _{x \rightarrow 0} \cos x=1$
(iii) $\lim\limits _{x \rightarrow 0} \sin x=0$

Some Standard Limit Functions to Remember:

1. $\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}$

2. $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

3. $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$

4. $\lim _{x \rightarrow 0} \frac{e^x-1}{x}=1$

5. $\lim _{x \rightarrow 0} \frac{a^x-1}{x}=\ln a$

6. $\lim _{x \rightarrow 0} \frac{\log (1+x)}{x}=1$

Limits of Polynomial and Rational Functions:

1. For a polynomial function:

$
\begin{gathered}
\lim _{x \rightarrow a} x=a \\
\lim _{x \rightarrow a} x^n=a^n
\end{gathered}
$

2. For a rational function $f(x)=\frac{g(x)}{h(x)}$ :

$\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} \frac{g(x)}{h(x)}=\frac{\lim _{x \rightarrow a} g(x)}{\lim _{x \rightarrow a} h(x)}=\frac{g(a)}{h(a)}, \quad \text {if } h(a) \neq 0$

Limits of Trigonometric Functions:

Theorem 2:

If $f(x)$ and $g(x)$ are two real-valued functions with the same domain such that

$f(x) \leq g(x)$

and if the limits $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ exist, then

$\lim _{x \rightarrow a} f(x) \leq \lim _{x \rightarrow a} g(x)$

Theorem 3: (Sandwich Theorem)

If $f(x), g(x)$, and $h(x)$ are real-valued functions with the same domain such that $f(x)≤g(x)≤h(x)$,
if $\lim _{x \rightarrow a} f(x)=l=\lim _{x \rightarrow a} h(x)$,
then $\lim _{x \rightarrow a} g(x)=l$

Derivatives

1. If $f$ is a real-valued function and $a$ belongs to its domain, the derivative of $f$ at $a$ is defined as:

$
f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}, \quad \text { if this limit exists. }
$
The derivative of $f$ at $a$ is denoted by $f^{\prime}(a)$.

2. Similarly, for a function $f(x)$, the derivative at $x$ is given by:

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}, \quad \text { if this limit exists. }$

Algebra of Derivatives of Functions

1. The derivative of the sum of two functions is equal to the sum of the individual derivatives of each function.

$\frac{d}{d x}[f(x)+g(x)]=\frac{d}{d x} f(x)+\frac{d}{d x} g(x)$

2. The derivative of the difference of two functions is equal to the derivative of the individual difference of each function.

$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x)$

3. The derivative of the product of two functions is equal to the derivative of the individual product of each function.

$\frac{d}{d x}[f(x) \cdot g(x)]=\frac{d}{d x} f(x) \cdot g(x)+f(x) \cdot \frac{d}{d x} g(x)$

4. The derivative of the quotient of two functions is equal to the quotient of the derivatives of each function.

$\frac{d}{d x}\left[\frac{f(x)}{g(x)}\right]=\frac{\frac{d}{d x} f(x) \cdot g(x)-f(x) \cdot \frac{d}{d x} g(x)}{[g(x)]^2}, \quad g(x) \neq 0$

Derivative of Polynomial and Trigonometric Functions:

If f(x) is said to be a polynomial function such that

$f(x)=a_n x^n+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_1 x+a_0$

Then the derivative of the function is given by:

$\frac{d}{d x} f(x)=n a_n x^{n-1}+(n-1) a_{n-1} x^{n-2}+(n-2) a_{n-2} x^{n-3}+\cdots+2 a_2 x+a_1$

Some standard Derivatives

1. $\frac{d}{d x}\left(x^n\right)=n x^{n-1}$
2. $\frac{d}{d x}(\sin x)=\cos x$
3. $\frac{d}{d x}(\cos x)=-\sin x$
4. $\frac{d}{d x}(\tan x)=\sec ^2 x$
5. $\frac{d}{d x}(\cot x)=-\operatorname{cosec}^2 x$
6. $\frac{d}{d x}(\sec x)=\sec x \cdot \tan x$
7. $\frac{d}{d x}(\operatorname{cosec} x)=-\operatorname{cosec} x \cdot \cot x$
8. $\frac{d}{d x}\left(a^x\right)=a^x \ln a$
9. $\frac{d}{d x}\left(e^x\right)=e^x$
10. $\frac{d}{d x}(\ln x)=\frac{1}{x}$