Class 11 RD Sharma Solutions – Chapter 29 Limits – Exercise 29.10 | Set 3

Read

Discuss

Evaluate the following limits:

Question 31. $\lim_{x\to0}\frac{e^{x+2}-e^x}{x}$

Solution:

We have,

= $\lim_{x\to0}\frac{e^{x+2}-e^x}{x}$

= $e^2\lim_{x\to0}\frac{e^x-1}{x}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= e² × log e

= e²

Question 32. $\lim_{x\to\frac{π}{2}}\frac{e^{cosx}-1}{cosx}$

Solution:

We have,

= $\lim_{x\to\frac{π}{2}}\frac{e^{cosx}-1}{cosx}$

Let x − π/2 = h. So, we get

= $\lim_{h\to0}\frac{e^{cos(\frac{π}{2}+h)}-1}{cos(\frac{π}{2}+h)}$

= $\lim_{sinh\to0}\frac{e^{-sinh}-1}{-sinh}$

We know $\lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we get,

= log e

= 1

Question 33. $\lim_{x\to0}\frac{e^{3+x}-sinx-e^3}{x}$

Solution:

We have,

= $\lim_{x\to0}\frac{e^{3+x}-sinx-e^3}{x}$

= $\lim_{x\to0}\frac{e^{3+x}-e^3}{x}-\lim_{x\to0}\frac{sinx}{x}$

= $e^3\lim_{x\to0}\frac{e^x-1}{x}-\lim_{x\to0}\frac{sinx}{x}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ and $lim_{x\to0}\frac{sinx}{x}=1$ . So, we have,

= e³ log e − 1

= e³ − 1

Question 34. $lim_{x\to0}\frac{e^x-x-1}{x}$

Solution:

We have,

= $lim_{x\to0}\frac{e^x-x-1}{x}$

= $lim_{x\to0}\frac{e^x-1}{x}-lim_{x\to0}\frac{x}{x}$

= $lim_{x\to0}(\frac{e^x-1}{x})-1$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= log e − 1

= 1 − 1

= 0

Question 35. $lim_{x\to0}\frac{e^{3x}-e^{2x}}{x}$

Solution:

We have,

= $lim_{x\to0}\frac{e^{3x}-e^{2x}}{x}$

= $lim_{x\to0}\frac{e^{3x}-1-(e^{2x}-1)}{x}$

= $lim_{x\to0}\frac{e^{3x}-1}{x}-lim_{x\to0}\frac{e^{2x}-1}{x}$

= $lim_{x\to0}3×(\frac{e^{3x}-1}{3x})-lim_{x\to0}2×(\frac{e^{2x}-1}{2x})$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= 3 log e − 2 log e

= 3 − 2

= 1

Question 36. $lim_{x\to0}\frac{e^{tanx}-1}{tanx}$

Solution:

We have,

= $lim_{x\to0}\frac{e^{tanx}-1}{tanx}$

= $lim_{tanx\to0}\frac{e^{tanx}-1}{tanx}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= log e

= 1

Question 37. $lim_{x\to0}\frac{e^{bx}-e^{ax}}{x}$ , 0 < a < b

Solution:

We have,

= $lim_{x\to0}\frac{e^{bx}-e^{ax}}{x}$

= $lim_{x\to0}\frac{e^{bx}-1-(e^{ax}-1)}{x}$

= $lim_{x\to0}\frac{e^{bx}-1}{x}-lim_{x\to0}\frac{e^{ax}-1}{x}$

= $lim_{x\to0}b×\frac{e^{bx}-1}{bx}-lim_{x\to0}a×\frac{e^{ax}-1}{ax}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= b log e − a log e

= b − a

Question 38. $lim_{x\to0}\frac{e^{tan}-1}{x}$

Solution:

We have,

= $lim_{x\to0}\frac{e^{tan}-1}{x}$

= $lim_{x\to0}\frac{e^{tan}-1}{tanx}×lim_{x\to0}\frac{tanx}{x}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ and $lim_{x\to0}\frac{tanx}{x}=1$ . So, we have,

= log e × 1

= 1

Question 39. $lim_{x\to0}\frac{e^x-e^{sinx}}{x-sinx}$

Solution:

We have,

= $lim_{x\to0}\frac{e^x-e^{sinx}}{x-sinx}$

= $lim_{x\to0}e^{sinx}\left[\frac{e^{x-sinx}-1}{x-sinx}\right]$

= $lim_{x\to0}e^{sinx}×lim_{x\to0}\left[\frac{e^{x-sinx}-1}{x-sinx}\right]$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= $lim_{x\to0}e^{sinx}×loge$

= e⁰

= 1

Question 40. $lim_{x\to0}\frac{3^{2+x}-9}{x}$

Solution:

We have,

= $lim_{x\to0}\frac{3^{2+x}-9}{x}$

= $lim_{x\to0}\frac{3^{2+x}-3^2}{x}$

= $3^2×lim_{x\to0}(\frac{3^x-1}{x})$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= 9 × log 3

= 9 log 3

Question 41. $lim_{x\to0}\frac{a^x-a^{-x}}{x}$

Solution:

We are given,

= $lim_{x\to0}\frac{a^x-a^{-x}}{x}$

= $lim_{x\to0}\frac{a^{2x}-1}{xa^x}$

= $lim_{x\to0}\frac{a^{2x}-1}{x}×lim_{x\to0}\frac{1}{a^x}$

= $lim_{x\to0}\frac{2(a^{2x}-1)}{2x}×lim_{x\to0}\frac{1}{a^x}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= $2loga×\frac{1}{a^0}$

= 2 log a

Question 42. $lim_{x\to0}\frac{x(e^x-1)}{1-cosx}$

Solution:

We have,

= $lim_{x\to0}\frac{x(e^x-1)}{1-cosx}$

= $lim_{x\to0}\frac{x(e^x-1)}{2sin^2\frac{x}{2}}$

= $lim_{x\to0}\frac{e^x-1}{2x}×lim_{x\to0}\frac{4}{\left(\frac{sin\frac{x}{2}}{\frac{x}{2}}\right)^2}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ and $lim_{x\to0}\frac{sinx}{x}=1$ . So, we have,

= $\frac{loge}{2}×4$

= 2

Question 43. $lim_{x\to\frac{π}{2}}\frac{2^{-cosx}-1}{xsin(x-\frac{π}{2})}$

Solution:

We have,

= $lim_{x\to\frac{π}{2}}\frac{2^{-cosx}-1}{xsin(x-\frac{π}{2})}$

= $lim_{x\to\frac{π}{2}}\frac{2^{-sin(\frac{π}{2}-x)}-1}{xsin(x-\frac{π}{2})}$

= $lim_{x\to\frac{π}{2}}\frac{2^{sin(x-\frac{π}{2})}-1}{xsin(x-\frac{π}{2})}$

= $lim_{x\to\frac{π}{2}}\frac{2^{sin(x-\frac{π}{2})}-1}{sin(x-\frac{π}{2})}×lim_{x\to\frac{π}{2}}\frac{1}{x}$

Let x− π/2 = h in first part. So, we get,

= $lim_{h\to0}\frac{2^{sinh}-1}{sinh}×lim_{x\to\frac{π}{2}}\frac{1}{x}$

We know, $lim_{x\to0}\frac{a^x-1}{x}=loga$ . So, we have,

= $log 2 ×\frac{2}{π}$

= $\frac{2log2}{π}$

= $\frac{log4}{π}$

Whether you're preparing for your first job interview or aiming to upskill in this ever-evolving tech landscape, GeeksforGeeks Courses are your key to success. We provide top-quality content at affordable prices, all geared towards accelerating your growth in a time-bound manner. Join the millions we've already empowered, and we're here to do the same for you. Don't miss out - check it out now!

Last Updated : 04 May, 2021

Class 11 RD Sharma Solutions - Chapter 11 Trigonometric Equations - Exercise 11.1

Class 12 RD Sharma Solutions - Chapter 28 The Straight Line in Space - Exercise 28.1 | Set 1

Class 11 RD Sharma Solutions – Chapter 29 Limits – Exercise 29.10 | Set 3

Evaluate the following limits:

Question 31.

Question 32.

Question 33.

Question 34.

Question 35.

Question 36.

Question 37., 0 < a < b

Question 38.

Question 39.

Question 40.

Question 41.

Question 42.

Question 43.

Please Login to comment...

Question 31. $\lim_{x\to0}\frac{e^{x+2}-e^x}{x}$

Question 32. $\lim_{x\to\frac{π}{2}}\frac{e^{cosx}-1}{cosx}$

Question 33. $\lim_{x\to0}\frac{e^{3+x}-sinx-e^3}{x}$

Question 34. $lim_{x\to0}\frac{e^x-x-1}{x}$

Question 35. $lim_{x\to0}\frac{e^{3x}-e^{2x}}{x}$

Question 36. $lim_{x\to0}\frac{e^{tanx}-1}{tanx}$

Question 37. $lim_{x\to0}\frac{e^{bx}-e^{ax}}{x}$ , 0 < a < b

Question 38. $lim_{x\to0}\frac{e^{tan}-1}{x}$

Question 39. $lim_{x\to0}\frac{e^x-e^{sinx}}{x-sinx}$

Question 40. $lim_{x\to0}\frac{3^{2+x}-9}{x}$

Question 41. $lim_{x\to0}\frac{a^x-a^{-x}}{x}$

Question 42. $lim_{x\to0}\frac{x(e^x-1)}{1-cosx}$

Question 43. $lim_{x\to\frac{π}{2}}\frac{2^{-cosx}-1}{xsin(x-\frac{π}{2})}$