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

Last updated date: 21st Jul 2024
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Vedantu’s Class 11 Chapter 29 - Limits RD Sharma Solutions

Free PDF download of RD Sharma Class 11 Solutions Chapter 29 - Limits Exercise 29.11 solved by Expert Mathematics Teachers on Vedantu. All Chapter 29 - Limits Ex 29.11 Questions with Solutions for RD Sharma Class 11 Math to help you to revise complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other Engineering entrance exams.

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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)

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

1. Does a Limit exist for zero?

Limit doesn’t exist for zero because the function has to approach the same value regardless of which direction x comes from. This doesn’t hold well for the function as x is approaching 0. Hence, the limit doesn’t exist for zero.  When the highest degree of both the numerator and the denominator are equal then one can use the coefficient s to determine the limit. In addition to this, when the highest degree of the numerator is larger than the highest degree of the denominator, the limit will be equal to infinity.

2. How to know if a limit is one sided?

A one-sided limit is a value that the given function approaches as the x-values approach the limit from one side only. Consider this for an example. f(x)=|x|/x will return -1 for the negative numbers, 1 for the positive numbers, and isn't defined for 0. The one-sided right limit of f at x=0 is 1, and the one-sided left limit at x=0 is -1. We differentiate both the numerator and the denominator of the rational function until the value of the limit is not in the form of 0/0.

3. Where is Calculus used in real life?

Limits and calculus are used as real-life approximations for calculating the derivatives. So, to make some calculations, engineers will try to approximate a function using small differences in the function and then try to calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals which we call limits in mathematics. So, Limits is a very important topic in engineering, especially relating to constructions.

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