 # Limits and Derivatives Class 11 Notes - Chapter 13

According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 12.

## What are the Limits?

l is called the limit of the function f(x) if the equation is given as x → a, f(x) → l, and this is symbolically written for all the limits, the function should assume at a given point x = a. x could approach a number in two ways, either from the left or from the right, i.e., all the values of x near a could be greater than a or could be less than a.

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### What is the Right-Hand Value?

In this type of limit, the Right-hand limit Value is referred to the situation in which f(x) gets dictated by values of f(x) when x tends to be from the right.

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### What is a Left-Hand Value?

In the case of the Left-hand limit, when x tends to be from the left, the value of f(x) gets dictated by values of f(x).

In this case, the right and left-hand limits are different, and hence we say that the limit of f(x) as x tends to zero does not exist (even though the function is defined at 0). This could also be followed in limits and continuity of values.

### Standards In Limits

The given are some limit standards:

• $$\begin{array}{l}{lim}_{x\rightarrow a}x^{n}-a^{n}/x-a=na^{n-1}\end{array}$$
• $$\begin{array}{l}{lim}_{x\rightarrow 0}\sin x/x=1\end{array}$$
• $$\begin{array}{l}{lim}_{x\rightarrow 0}1-\cos x/x=0\end{array}$$

### Derivative of the Function

If we are referring to a function f at a, then its equation is defined by

$$\begin{array}{l}f'(a)=lim_{h\rightarrow 0}f(a+h)-f(a)/h\end{array}$$

Similarly, the derivation of a function f at x is :

$$\begin{array}{l}f'(x)=df(x)/dx=lim_{h\rightarrow 0}f(x+h)-f(x)/h\end{array}$$

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### Standards In Derivatives

The given are some derivative standards:

• $$\begin{array}{l}d/dx(x^{n})=nx^{n-1}\end{array}$$
• $$\begin{array}{l}d/dx(\sin x)=\cos x\end{array}$$
• $$\begin{array}{l}d/dx(\cos x)=-\sin x\end{array}$$

### Important Questions

1. $$\begin{array}{l}\lim_{x\rightarrow 0}\sin (\pi -x)/\pi (\pi -x)\end{array}$$
2. $$\begin{array}{l}\lim_{x\rightarrow 0}\sin (\pi -x)/\pi (\pi -x)\end{array}$$
3.
$$\begin{array}{l}\lim_{x\rightarrow 0}cos x/\pi -x\end{array}$$
4. $$\begin{array}{l}\lim_{x\rightarrow 0}ax+x\cos x/b\sin x\end{array}$$
5. $$\begin{array}{l}\lim_{x\rightarrow 0}x\sec x\end{array}$$
6. $$\begin{array}{l}\lim_{x\rightarrow \pi }\tan 2x/x-\pi /2\end{array}$$

## Frequently Asked Questions on CBSE Class 11 Maths Notes Chapter 13 Limits and Derivatives

Q1

### What are the uses of Calculus?

Calculus has applications in various fields, such as engineering, medicine, biological research, economics, architecture, space science, electronics, statistics, and pharmacology.

Q2

### What is the limit formula?

The limit formula is used to calculate the derivative of a function.

Q3

### What is a derivative?

In mathematics (differential calculus), the derivative is a way to show the instantaneous rate of change.