Limits and Derivatives

Limits and derivatives are incredibly important topics in mathematics that have applications not only in math but also in other fields. Velocity, density, current, power and temperature gradient in physics, rate of reaction and compressibility in chemistry, rate of growth and blood velocity in biology, marginal cost and marginal profit in economics, rate of heat flow in geology, rate of improvement in performance in psychology - these are all cases of a single mathematical concept, the derivative. Limit plays a major role in calculus in terms of arriving at a value. A limit is a value that a function approaches as an input and creates some value in mathematics. Limits are used to define integrals, derivatives, and continuity in calculus and mathematical analysis.

This is an illustration of the fact that part of the power of mathematics lies in its abstractness. A single abstract mathematical concept such as derivatives can have different interpretations in each of the sciences.

This article is about the concept of limits and derivatives class 11. Limits and derivatives chapter is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, BCECE, and more.

Limits and Derivatives

In calculus, limits and derivatives are crucial for understanding the behavior of functions as they approach specific points. Mastering limits is key to grasping fundamental calculus concepts. They help analyze changes, optimize processes, and predict trends in fields like engineering, physics, and economics. Limits allow us to study functions near critical points, even if the function is not defined at those points.

What is a Limit?

A limit describes the value that a function $f(x)$ approaches as the variable $x$ approaches a particular point $a$.

Let $I$ be an open interval containing $x_0\in$. Let $f:I\to \mathbb{R}$. Then we say that the limit of $f(x)$ is $L$, as $x$ approaches $x_0$ [symbollically written as $\underset{x\to x_0}{lim}f(x)=L$ ], if, whenever $x$ becomes sufficiently close to $x_0$ from either side with $x\ne x_0,f(x)$ gets sufficiently close to $L$.

In formulas, a limit of a function is usually written as $\underset{x\to c}{lim}f(x)=L$, and is read as the limit of $f$ of $x$ as $x$ approaches $c$ equals $L$.

Let's consider the function $\mathrm{f}(x)=x^2$

Observe that as $x$ takes values very close to $0$, the value of $f(x)$ is also close to $0$. (See graph below)

We can also interpret it in another way. If we input the values of $x$ which tend to approach $0$ (meaning close to $0$, either just smaller than $0$ or just larger than $0$ ), the value of $f(x)$ will tend to approach $0$(meaning close to $0$, either just smaller than $0$ or just larger than $0$).

Then we can say that, $\underset{x\to 0}{lim}f(x)=0$

Similarly, when $x$ approaches $2$, the value of $f(x)$ approaches $4$, i.e. $\underset{x\to 2}{lim}f(x)=4$ or $\underset{x\to 2}{lim}{x}^2=4$.

In general, as $x\to a$ , $f(x)\to l$, then $l$ is called the limit of the function $\mathrm{f}(\mathrm{x})$, which is symbolically written as $\underset{x\to a}{lim}f(x)=l$.

Even if the function does not exist at $x=a$, still the limit can exist at that point as the limit is concerned only about the points close to $x=a$ and not at $x=a$ itself.

Some Properties of Limits:

• $\underset{\mathrm{x}\to \mathrm{a}}{lim}\mathrm{c}=\mathrm{c}$, where $c$ is a constant quantity.
• The value of $\underset{\mathrm{x}\to \mathrm{a}}{lim}x=\mathrm{a}$.
• Value of $\underset{x\to a}{lim}(bx+c)=ba+c$.
• $\underset{x\to a}{lim}{x}^n=a^n$, if $n$ is a positive integer.

Left-Hand Limits

Left-Hand Limit $\underset{x\to a^{-}}{lim}f(x)$: The limit of $f(x)$ as $x$ approaches $c$ from the left - that is, through numbers smaller than $c$. The left-hand limit of a function $f(x)$ as $x$ approaches a from the left is denoted by $\underset{x\to a^{-}}{lim}f(x)=LHL$

Right-Hand Limits

Right-Hand Limit $\underset{x\to a^{+}}{lim}f(x)$ The limit of $f(x)$ as $x$ approaches $c$ from the right - that is, through numbers bigger than $c$. The right-hand limit of a function $f(x)$ as $x$ approaches a from the right is denoted by $\underset{x\to a^{+}}{lim}f(x)=RHL$

Now consider a function, $f(x)=\frac{|x|}{x}$
Let's check the behavior of $f(x)$ in the neighborhood of $x=0$

$\mathrm{LHL}=\underset{\mathrm{x}\to 0^{-}}{lim}\frac{|\mathrm{x}|}{\mathrm{x}}$

As $x$ is just less than $0$, we can replace it by $(0-h)$, where $h$ is positive and very close to $0$

$=\underset{\mathrm{h}\to 0^{+}}{lim}\frac{0-\mathrm{h}\mid }{0-\mathrm{h}}$

So, we have

$\begin{array}{rl}& =\underset{h\to 0^{+}}{lim}\frac{|-h|}{-h}\\ & =\underset{h\to 0^{+}}{lim}\frac{h}{-h}\\ & =-1\end{array}$
$\begin{array}{rl}\mathrm{RHL}& =\underset{\mathrm{x}\to 0^{+}}{lim}\frac{|x|}{\mathrm{x}}\\ & =\underset{\mathrm{h}\to 0^{+}}{lim}\frac{|0+\mathrm{h}|}{0+\mathrm{h}}\\ & =\underset{\mathrm{h}\to 0^{+}}{lim}\frac{\mathrm{h}}{\mathrm{h}}=1\end{array}$
Here, we have RHL $\ne$ LHL

Existence of a limit of a function

From the above example, we can define the existence of limit

The limit of a function $f(x)$ at $x=a$ exists if $\underset{x\to a^{-}}{lim}f(x)=\underset{x\to a^{-}}{lim}f(x)$

i.e., $LHL=RHL$ at $x=a$

Also, notice that the limit of a function can exist even when $f(x)$ is not defined at $x=a$.

Derivatives

The value obtained after differentiating a function is called the derivative. Let $f$ be defined on an open interval $I\subseteq$ containing the point $x_0$, and suppose that $\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x_0+\mathrm{\Delta }x)-f\left(x_0\right)}{\mathrm{\Delta }x}$ exists. Then $f$ is said to be differentiable at $x_0$ and the derivative of $f$ at $x_0$, denoted by $f^{\prime }\left(x_0\right)$, is given by

$f^{\prime }\left(x_0\right)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x_0+\mathrm{\Delta }x)-f\left(x_0\right)}{\mathrm{\Delta }x}$

For all $x$ for which this limit exists,
$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$ is a function of $x$.

In addition to $f^{\prime }(x)$, other notations are used to denote the derivative of $y=f(x)$. The most common notations are $f^{\prime }(x),\frac{dy}{dx},y^{\prime },\frac{d}{dx}[f(x)],D_x[y]$ or $Dy$ or $y_1$. Here $\frac{d}{dx}$ or $D$ is the differential operator.

Limits and Derivative Class 11 Formulas

Limits Formulas

Operation on Limits

The operation on limits include sum, difference, constant multiplication, product, quotient, power and composition of functions.

Sum law for limits : $\underset{x\to a}{lim}(f(x)+g(x))=\underset{x\to a}{lim}f(x)+\underset{x\to a}{lim}g(x)=L+M$

Difference law for limits : $\underset{x\to a}{lim}(f(x)-g(x))=\underset{x\to a}{lim}f(x)-\underset{x\to a}{lim}g(x)=L-M$

Constant multiple law for limits : $\underset{x\to a}{lim}cf(x)=c\cdot \underset{x\to a}{lim}f(x)=cL$

Product law for limits : $\underset{x\to a}{lim}(f(x)\cdot g(x))=\underset{x\to a}{lim}f(x)\cdot \underset{x\to a}{lim}g(x)=L\cdot M$

Quotient law for limits : $\underset{x\to a}{lim}\frac{f(x)}{g(x)}=\frac{\underset{x\to a}{lim}f(x)}{\underset{x\to a}{lim}g(x)}=\frac{L}{M}$ for $M\ne 0$

Power law for limits : $\underset{x\to a}{lim}(f(x){)}^n={(\underset{x\to a}{lim}f(x))}^n=L^n$ for every positive integer $n$

Composition law of limit: $\underset{x\to a}{lim}(f\circ g)(x)=f(\underset{x\to a}{lim}g(x))=f(M)$, only if $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{g}(\mathrm{x})=\mathrm{M}$.

If $\underset{x\to a}{lim}f(x)=+\mathrm{\infty }$ or $-\mathrm{\infty }$, then $\underset{x\to a}{lim}\frac{1}{f(x)}=0$

Indeterminate Forms

Indeterminate forms arise in limits when the standard limit rules yield expressions that do not directly lead to a specific value. If we directly substitute $x=a$ in $f(x)$ while evaluating $\underset{x\to a}{lim}f(x)$ and will get one of the seven following forms $\frac{0}{0},\frac{\mathrm{\infty }}{\mathrm{\infty }},\mathrm{\infty }-\mathrm{\infty },1^{\mathrm{\infty }},0^0,{\mathrm{\infty }}^0,\mathrm{\infty }\times 0$ then it is called indeterminate form.

Example: $\underset{x\to 2}{lim}\frac{x^2-4}{x-2}=\frac{0}{0}$ indeterminate form.

L'Hospital's Rule

L'Hospital's Rule states that, if $\underset{x\to a}{lim}\frac{f(x)}{g(x)}$ is of $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ form , then differentiate numerator and denominator till this intermediate form is removed. $\underset{x\to a}{lim}\frac{f(x)}{g(x)}=\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$

But, if we again get the indeterminate form $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}\underset{x\to a}{lim}\frac{f(x)}{g(x)}=\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}=\underset{x\to a}{lim}\frac{f^{\prime \prime }(x)}{g^{\prime \prime }(x)}$ (so we differentiate numerator and denominator again)
This process is continued till the indeterminate form $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ is removed.

Example:

Evaluate $\underset{x\to 0}{lim}\frac{\sin x}{x}$

$\underset{x\to 0}{lim}\frac{\sin x}{x}=\frac{0}{0}$

But by L'Hospital rule, $\underset{x\to 0}{lim}\frac{\sin x}{x}=\underset{x\to 0}{lim}\frac{\cos x}{1}=1$

Derivatives Formula

Derivatives of Some Basic Functions

The differentiation of some basic functions are

1. $\frac{d}{dx}($ constant $)=0$

2. $\frac{d}{dx}\left({\mathbf{x}}^{\mathbf{n}}\right)=\mathbf{n}{\mathbf{x}}^{\mathbf{n}-1}$

3. $\frac{d}{dx}\left({\mathbf{a}}^{\mathrm{x}}\right)={\mathbf{a}}^{\mathrm{x}}{\log }_{\mathrm{e}}\mathbf{a}$

4. $\frac{d}{dx}\left({\mathrm{e}}^{\mathrm{x}}\right)={\mathrm{e}}^{\mathrm{x}}{\log }_{\mathrm{e}}\mathrm{e}={\mathrm{e}}^{\mathrm{x}}$

5. $\frac{d}{dx}({\log }_{\mathrm{e}}|\mathbf{x}|)=\frac{\mathbf{1}}{\mathbf{x}},\mathbf{x}\ne 0$

6. $\frac{d}{dx}({\log }_{\mathbf{a}}|\mathbf{x}|)=\frac{1}{\mathbf{x}{\log }_{\mathrm{e}}\mathbf{a}},\mathbf{x}\ne 0$

7. $\frac{d}{dx}(\sin (\mathbf{x}))=\cos (\mathbf{x})$

8. $\frac{d}{dx}(\cos (\mathbf{x}))=-\sin (\mathbf{x})$

9. $\frac{d}{dx}(\tan (\mathbf{x}))={\sec }^2(\mathbf{x})$

10. $\frac{d}{dx}(\cot (\mathbf{x}))=-{\csc }^2(\mathbf{x})$

11. $\frac{d}{dx}(\sec (\mathbf{x}))=\sec (\mathbf{x})\tan (\mathbf{x})$

12. $\frac{d}{dx}(\csc (\mathbf{x}))=-\csc (\mathbf{x})\cot (\mathbf{x})$

Rules of Differentiation

The important rules of differentiation are

• Power Rule
• Sum and Difference Rule
• Product Rule
• Quotient Rule
• Chain Rule

Let $f(x)$ and $g(x)$ be differentiable functions and $k$ be a constant. Then each of the following rules holds

Sum Rule

The derivative of the sum of a function $f$ and a function $g$ is the same as the sum of the derivative of $f$ and the derivative of $g$.

$\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x))$
In general,

$\frac{d}{dx}(f(x)+g(x)+h(x)+\dots \dots )=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x))+\frac{d}{dx}(h(x))+\dots \dots$

Difference Rule

The derivative of the difference of a function $f$ and $a$ function $g$ is the same as the difference of the derivative of $f$ and the derivative of $g$.

$\begin{array}{rl}& \frac{d}{dx}(f(x)-g(x))=\frac{d}{dx}(f(x))-\frac{d}{dx}(g(x))\\ & \frac{d}{dx}(f(x)-g(x)-h(x)-\dots \dots )=\frac{d}{dx}(f(x))-\frac{d}{dx}(g(x))-\frac{d}{dx}(h(x))-\dots \dots \end{array}$

Constant Multiple Rule

The derivative of a constant $k$ multiplied by a function $f$ is the same as the constant multiplied by the derivative of $f$

$\frac{d}{dx}(kf(x))=k\frac{d}{dx}(f(x))$

Product rule

Let $f(x)$ and $g(x)$ be differentiable functions. Then,

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

This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.

Extending the Product Rule

If three functions are involved, i.e let $k(x)=f(x)\cdot g(x)\cdot h(x)$
Let us have a function $\mathrm{k}(\mathrm{x})$ as the product of the function $\mathrm{f}(\mathrm{x}),\mathrm{g}(\mathrm{x})$ and $\mathrm{h}(\mathrm{x})$. That is, $k(x)=(f(x)\cdot g(x))\cdot h(x)$. Thus,

$k^{\prime }(x)=\frac{d}{dx}(f(x)g(x))\cdot h(x)+\frac{d}{dx}(h(x))\cdot (f(x)g(x))$

[By applying the product rule to the product of $f(x)g(x)$ and $h(x)$.]

$\begin{array}{rl}& =(f^{\prime }(x)g(x)+g^{\prime }(x)f(x))h(x)+h^{\prime }(x)f(x)g(x)\\ & =f^{\prime }(x)g(x)h(x)+f(x)g^{\prime }(x)h(x)+f(x)g(x)h^{\prime }(x)\end{array}$

Quotient Rule

Let $f(x)$ and $g(x)$ be differentiable functions. Then

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x)\cdot \frac{d}{dx}(f(x))-f(x)\cdot \frac{d}{dx}(g(x))}{(g(x){)}^2}$
OR
if $h(x)=\frac{f(x)}{g(x)}$, then $h^{\prime }(x)=\frac{f^{\prime }(x)g(x)-g^{\prime }(x)f(x)}{(g(x){)}^2}$
As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives.

Chain Rule

If $u(x)$ and $v(x)$ are differentiable funcitons, then $uov(x)$ or $u[v(x){]}_{\text{is also differentiable. }}$
If $y=uov(x)=u[v(x)]$, then

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}u\{v(x)\}}{\mathrm{d}\{v(x)\}}\times \frac{\mathrm{d}}{\mathrm{d}x}v(x)$

is known as the chain rule. Or,

$\text{ If }y=f(u)\text{ and }u=g(x)\text{, then }\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\cdot \frac{\mathrm{d}u}{\mathrm{d}x}$
The chain rule can be extended as follows
If $y=[\mathrm{uovow}(x)]=u[v\{w(x)\}]$, then

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}[u[v\{w(x)\}]}{\mathrm{d}v\{w(x)\}}\times \frac{\mathrm{d}[v\{w(x)\}]}{\mathrm{d}w(x)}\times \frac{\mathrm{d}[w(x)]}{\mathrm{d}x}$

List of Topics According to NCERT/JEE MAIN

Importance of Limits and Derivatives Class 11

Limits and Derivatives have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

How to Study Limits and Derivatives?

Start preparing by understanding and practicing the concept of limits. Try to be clear on concepts like formulas for differentiation and differentiation rules. Practice many problems from each topic for better understanding.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

NCERT Notes Subject wise link:

• NCERT Notes for class 12 Physics.
• NCERT Notes for class 12 Chemistry.
• NCERT Notes for class 12 Mathematics.
• NCERT Notes for class 12 Biology

Important Books for Limits and Derivatives

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Amit M Aggarwal's differential and integral calculus or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Limits and Derivatives are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

• NCERT solutions for class 11 Physics.

• NCERT solutions for class 11 Chemistry.

• NCERT solutions for class 11 Mathematics.

• NCERT solutions for class 11 Biology.

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