Limits and Derivatives

What is a Limit?

A limit describes the value that a function approaches as the input gets closer to a specific point. Even if the function is not defined at that point, the limit can still exist by observing the behavior near it.

Mathematically, the limit of a function $f(x)$ as $x$ approaches $a$ is written as:

$\lim_{x \to a} f(x) = L$

This means that as $x$ gets closer to $a$, the value of $f(x)$ approaches $L$.

Example

Consider the function $f(x) = x^2$.

• As $x$ approaches $0$, $f(x)$ approaches $0$.

  So, $\lim_{x \to 0} x^2 = 0$.

• As $x$ approaches $2$, $f(x)$ approaches $4$.

  So, $\lim_{x \to 2} x^2 = 4$.

Thus, in general, if $x$ approaches $a$, then $f(x)$ approaches some value $l$:

$\lim_{x \to a} f(x) = l$

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.

Properties of Limits

The following properties are used frequently while solving problems in limits and derivatives class 11:

1. Limit of a constant
   If $c$ is a constant, then $\lim_{x \to a} c = c$
   The limit of a constant is the constant itself.

2. Limit of the identity function
   $\lim_{x \to a} x = a$
   As $x$ approaches $a$, the value of $x$ itself approaches $a$.

3. Sum of limits
   $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
   The limit of the sum is the sum of the limits.

4. Difference of limits
   $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
   The limit of the difference is the difference of the limits.

5. Constant multiple of a limit
   $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$
   A constant factor can be pulled out of the limit.

6. Product of limits
   $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
   The limit of a product is the product of the limits.

7. Quotient of limits
   If $\lim_{x \to a} g(x) \neq 0$, then
   $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$
   The limit of a quotient is the quotient of the limits.

8. Power of a limit
   If $n$ is a positive integer,
   $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
   The limit of a function raised to a power is the power of the limit.

9. Root of a limit
   If $n$ is a positive integer and the limit is defined,
   $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$
   The limit of a root is the root of the limit.

10. Polynomial and rational functions
    Limits of polynomial and rational functions can be evaluated by direct substitution if the function is defined at that point.

Left-Hand and Right-Hand Limits – Class 11 Limits and Derivatives

When learning limits and derivatives class 11, it’s important to know how a function behaves when the input gets close to a particular value from different directions. This is where left-hand limits and right-hand limits are used. They help us understand the behavior of functions as the variable approaches a point from either smaller or larger values.

Left-Hand Limit

The left-hand limit, written as $ \lim_{x \to a^{-}} f(x) $, describes how the function $f(x)$ behaves when $x$ gets closer to $a$ from values smaller than $a$. In other words, we are observing what happens to the function when $x$ approaches $a$ from the left side of the number line.

For example, if we write $ \lim_{x \to a^{-}} f(x) = LHL $, it means that as $x$ gets closer to $a$ from the left side, the value of $f(x)$ approaches $LHL$.

Right-Hand Limit

The right-hand limit, written as $ \lim_{x \to a^{+}} f(x) $, describes how the function $f(x)$ behaves when $x$ approaches $a$ from values greater than $a$. This helps us study the behavior of the function from the right side of $a$.

If we write $ \lim_{x \to a^{+}} f(x) = RHL $, it means that as $x$ gets closer to $a$ from the right side, the value of $f(x)$ approaches $RHL$.

Example – Let’s understand this with an example, often seen in class 11 maths limits and derivatives miscellaneous exercise.

Consider the function $ f(x) = \frac{|x|}{x} $. We want to find how this function behaves near $x = 0$ by calculating both the left-hand limit and the right-hand limit.

Left-Hand Limit

We calculate the left-hand limit as:

$ LHL = \lim_{x \to 0^{-}} \frac{|x|}{x} $

Since $x$ is slightly less than $0$, we can write $x = 0 - h$, where $h > 0$ and very small. Therefore,

$ = \lim_{h \to 0^{+}} \frac{|0 - h|}{0 - h} $

$ = \lim_{h \to 0^{+}} \frac{h}{-h} $

$ = -1 $

So, the left-hand limit at $x = 0$ is $-1$.

Right-Hand Limit

Now, we calculate the right-hand limit as:

$ RHL = \lim_{x \to 0^{+}} \frac{|x|}{x} $

Since $x$ is slightly greater than $0$, we write $x = 0 + h$, where $h > 0$ and very small:

$ = \lim_{h \to 0^{+}} \frac{|0 + h|}{0 + h} $

$ = \lim_{h \to 0^{+}} \frac{h}{h} $

$ = 1 $

So, the right-hand limit at $x = 0$ is $1$.

Existence of a Limit – Class 11 Limits and Derivatives

From the above example, we see that the left-hand limit is $-1$ and the right-hand limit is $1$. Since they are not equal, the limit at $x = 0$ does not exist.

In general, the limit of a function at $x = a$ exists only if the left-hand limit and right-hand limit are equal. That is, $ \lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) $.

Even if the function is not defined at $x = a$, the limit can still exist if the values from the left and right sides approach the same number. This concept is very useful in class 11 maths limits and derivatives, as it helps us analyze functions at points where they may not be defined but behave consistently near that point.

Derivatives

In real life, we often need to know how quickly something is changing. For example, if you are driving, you may want to know how fast your speed is increasing at a particular moment. This rate of change is what we find using derivatives in class 11 limits and derivatives. The derivative helps us understand how a function changes at any point and is widely used in science, engineering, and economics.

What is a Derivative?

The derivative of a function gives the rate at which the function’s value changes as the input changes. Suppose a function $f$ is defined on an interval that contains a point $x_0$, and the following limit exists:

$ \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} $

Then, the function is said to be differentiable at $x_0$, and the derivative of $f$ at $x_0$ is written as $f^{\prime}(x_0)$ and is defined by:

$ f^{\prime}(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} $

This means that the derivative measures how $y$ (which is $f(x)$) changes when $x$ is changed by a small amount $ \Delta x $.

Derivative at All Points

For all values of $x$ where this limit exists, the derivative is written as:

$ f^{\prime}(x) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $

This means that the derivative is itself a function of $x$, telling us how $f(x)$ changes at any point.

Notations of Derivatives

There are several common ways to denote the derivative of $y = f(x)$. Some of the widely used notations in class 11 maths limits and derivatives include:

• $f^{\prime}(x)$
• $\frac{dy}{dx}$
• $y^{\prime}$
• $\frac{d}{dx}[f(x)]$
• $D_x[y]$
• $Dy$
• $y_1$

Here, the symbols $\frac{d}{dx}$ or $D$ represent the differential operator, which is used to calculate derivatives.

Limits and Derivative Class 11 Formulas

Operation on Limits

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

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

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

Constant multiple law for limits : $\lim\limits _{x \rightarrow a} c f(x)=c \cdot \lim\limits _{x \rightarrow a} f(x)=c L$

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

Quotient law for limits : $\lim\limits _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim\limits _{x \rightarrow a} f(x)}{\lim\limits _{x \rightarrow a} g(x)}=\frac{L}{M}$ for $M \neq 0$

Power law for limits : $\lim\limits _{x \rightarrow a}(f(x))^n=\left(\lim\limits _{x \rightarrow a} f(x)\right)^n=L^n$ for every positive integer $n$

Composition law of limit: $\lim\limits _{x \rightarrow a}(f \circ g)(x)=f\left(\lim\limits _{x \rightarrow a} g(x)\right)=f(M)$, only if $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{g}(\mathrm{x})=\mathrm{M}$.

If $\lim\limits _{x \rightarrow a} f(x)=+\infty$ or $-\infty$, then $\lim\limits _{x \rightarrow a} \frac{1}{f(x)}=0$

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 $\lim\limits _{x \rightarrow a} f(x)$ and will get one of the seven following forms $\frac{0}{0}, \frac{\infty}{\infty}, \infty-\infty, 1^{\infty}, 0^0, \infty^0, \infty \times 0$ then it is called indeterminate form.

Example: $\lim\limits _{x \rightarrow 2} \frac{x^2-4}{x-2}=\frac{0}{0}$ indeterminate form.

L'Hospital's Rule

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

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

Example:

Evaluate $\lim\limits _{x \rightarrow 0} \frac{\sin x}{x}$

$
\lim\limits _{x \rightarrow 0} \frac{\sin x}{x}= \frac{0}{0}
$

But by L'Hospital rule, $\lim\limits _{x \rightarrow 0} \frac{\sin x}{x}=\lim\limits _{x \rightarrow 0} \frac{\cos x}{1}=1$

Derivatives of Some Basic Functions

The differentiation of some basic functions are

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

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

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

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

5. $\frac{d}{d x}\left(\log _{\mathrm{e}}|\mathbf{x}|\right)=\frac{\mathbf{1}}{\mathbf{x}}, \quad \mathbf{x} \neq 0$

6. $\quad \frac{d}{d x}\left(\log _{\mathbf{a}}|\mathbf{x}|\right)=\frac{1}{\mathbf{x} \log _{\mathrm{e}} \mathbf{a}}, \quad \mathbf{x} \neq 0$

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

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

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

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

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

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

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

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

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{aligned}
& \frac{d}{d x}(f(x)-g(x))=\frac{d}{d x}(f(x))-\frac{d}{d x}(g(x)) \\
& \frac{d}{d x}(f(x)-g(x)-h(x)-\ldots \ldots)=\frac{d}{d x}(f(x))-\frac{d}{d x}(g(x))-\frac{d}{d x}(h(x))-\ldots \ldots
\end{aligned}
$

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

Product rule

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

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

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

$
\begin{aligned}
& =\left(f^{\prime}(x) g(x)+g^{\prime}(x) f(x)\right) 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{aligned}
$

Quotient Rule

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

$
\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x) \cdot \frac{d}{d x}(f(x))-f(x) \cdot \frac{d}{d x}(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 $u o v(x)$ or $u[v(x)]_{\text {is also differentiable. }}$
If $y=u o v(x)=u[v(x)]$, 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 $y=[\operatorname{uovow}(x)]=u[v\{w(x)\}]$, 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 related to Limits and Derivatives according to NCERT/JEE MAIN

This section covers all the important topics from limits and derivatives class 11 as per NCERT and JEE Main, helping students focus on key areas for better preparation.

Important Books and Resources for Limits and Derivatives

Here you will find the best books and study materials that explain limits and derivatives clearly and provide practice questions aligned with the class 11 maths limits and derivatives syllabus.

NCERT Resources

This part lists the essential NCERT resources for studying limits and derivatives, which form the foundation for understanding concepts and solving problems effectively.

• NCERT Maths Notes for Class 11th Chapter 13 - Limits and Derivatives

• NCERT Maths Solutions for Class 11th Chapter 13 - Limits and Derivatives

• NCERT Maths Exemplar Solutions for Class 11th Chapter 13 - Limits and Derivatives

NCERT Subjectwise Resources

In this section, you will get subject-specific NCERT resources that offer detailed explanations, examples, and solved questions related to class 11.

Practice Questions based on Limits and Derivatives

This section provides practice questions based on limits and derivatives, helping students strengthen their problem-solving skills and prepare for exams like JEE Main.