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. $\lim_{x\rightarrow \pi }\tan 2x/x-\pi /2$