 # Limits and Derivatives Class 11 Notes - Chapter 13

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

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### What is a Left-Hand Value?

In the case of Left-hand limit, when x tends to 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’s standards:

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

### Derivative of the Function

If we are referring to a function f at a, then it’s equation is defined by

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

Similarly the derivation of a function f at x is :

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

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### Standards In Derivatives

The given are some derivative’s standards:

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

### Important Questions

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