RD Sharma Class 11 Solutions Chapter 29 - Limits (Ex 29.11) Exercise 29.11 - Free PDF

In Mathematics, limits are defined as the values that a given function approaches the output values for the given input values. Limits play a huge role in the topics of  calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It concerns the behavior of the given function at some particular points. 

Limits Formula

Let y = f(x) be a function in x. If at a specific point x = a, f(x) takes the indeterminate form, then we consider the values of the function which is very near to a. If these values tend to some specific definite unique number as x tends to a, then that unique number which we obtained is called the limit of f(x) at x = a.

Limits Formulae

Following are some of the most important formulae in limits chapter

1. \[lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x-a}=na^{(n-1)}\] , for all real values of n.

2. \[lim_{\theta \rightarrow 0}\frac{sin\theta }{\theta }=1\]

3. \[lim_{\theta \rightarrow 0}\frac{tan\theta }{\theta }=1\]

4. \[lim_{\theta \rightarrow 0}\frac{1-cos\theta }{\theta }=0\]

5. \[lim_{\theta \rightarrow 0}\; cos\theta =1\]

6. \[lim_{x\rightarrow 0}\: e^{x}=1\]

7. \[lim_{x\rightarrow 0}\frac{e^{x}-1}{x}=1\]

8. \[lim_{x\rightarrow \infty}(1+\frac{1}{x})^{x}=e\]

Limits of Functions and Continuity

Limits of the function and continuity of the function are related to each other very closely. Functions can be continuous or discontinuous. For a function to be a continuous function, if there are small changes in the input of the function then there must be small changes in the output. 

Limit of a Polynomial Function

Consider the following polynomial function, f(x) = a0 + a1x + a2x2 + … + anxn. Here, a0, a1, … , and which are all constants. At any given point, say x = a, the limit of this given polynomial function is

\[lim_{x\rightarrow a}f(x)=lim_{x\rightarrow a}[a_{0} +a_{1}x + a_{2}x_{2} +\cdots +a_{n}x_{n}]=lim_{x\rightarrow a}a_{0} +a_{1}lim_{x\rightarrow a}x+a_{2}lim_{x\rightarrow a}x_{2}+\cdots +a_{n}lim_{x\rightarrow a}x_{n}\] 

or,\[lim_{x\rightarrow a}=a_{0}+a_{1}a+a_{2}a_{2}+\cdots +a_{n}a_{n}=f(a)\]

Limit of a Rational Function

The limit of any rational function which is of the type p(x) / q(x), where q(x) ≠ 0 and p(x) and q(x) are polynomial functions, is

\[lim_{x\rightarrow a}[p(x)/q(x)]=[lim_{x\rightarrow a}p(x)]/[lim_{x\rightarrow a}q(x)]=p(a)/q(a)\]

The first step to find the limit of a rational function is to check whether it is reduced to the form 0/0 at some particular point. If the case is so, then, some adjustment is to be done so that one can easily calculate the value of the limit. This can be done in two ways.

• Canceling the factor which causes the limit to be of the form 0/0 here.

Assume the function, \[f(x)=(x^{2}+4x+4)/(x^{2}-4)\]. On taking limit over it for x = −2, the function is of the form 0/0,

\[lim_{x\rightarrow 2} f(x) = lim_{x\rightarrow 2}[(x^{2}+4x+4)/(x^{2}-4)]=lim_{x\rightarrow 2}[(x+2)^{2}/(x-2)(x+2)=0/-4(\neq 0/0)]=0\]

•  Applying the L – Hospital’s Rule here.

Assume a function, f(x) = sin x/x. Taking the limit over it for x = 0, the function is of the form 0/0. On taking the differentiation of both sin x and x with respect to x in the limit, \[lim_{x\rightarrow 0} sinx/x\] reduces to \[lim_{x\rightarrow 0}cosx/1=1\]. (cos 0 = 1)