Class 11 Maths Chapter 12 Limits and Derivatives – Exercise 12.1 NCERT Solutions

Introduction

Exercise 12.1 introduces the concept of limits in calculus. You will learn how to evaluate limits of algebraic, trigonometric, and rational functions. This exercise builds the foundation for derivatives and continuity in higher mathematics.

Key Concepts

1. Limit Definition:

$$\underset{x\to a}{\lim }f(x)=L$$

if $f(x)$ approaches $L$ as $x$ approaches $a$.

2. Important Standard Limits:

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$$

$$\underset{x\to 0}{\lim }\frac{\tan x}{x}=1$$

$$\underset{x\to 0}{\lim }\frac{1-\cos x}{x^2}=\frac{1}{2}$$

$$\underset{x\to \mathrm{\infty }}{\lim }{(1+\frac{1}{x})}^x=e$$

3. Algebra of Limits:

   • $\lim (f+g)=\lim f+\lim g$

   • $\lim (f\cdot g)=\lim f\cdot \lim g$

   • $\lim \left(\frac{f}{g}\right)=\frac{\lim f}{\lim g}$, if $\lim g\mathrm{\ne }0$.

NCERT Questions with Solutions (10)

Q1. Evaluate ${\lim }_{x\to 0}\frac{\sin x}{x}$.

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$$

Q2. Evaluate ${\lim }_{x\to 0}\frac{\tan x}{x}$.

$$\underset{x\to 0}{\lim }\frac{\tan x}{x}=1$$

Q3. Evaluate ${\lim }_{x\to 0}\frac{1-\cos x}{x^2}$.

$$\underset{x\to 0}{\lim }\frac{1-\cos x}{x^2}=\frac{1}{2}$$

Q4. Evaluate ${\lim }_{x\to 0}\frac{\sin 3x}{x}$.

$$\underset{x\to 0}{\lim }\frac{\sin 3x}{x}=3$$

Q5. Evaluate ${\lim }_{x\to 0}\frac{\tan 5x}{x}$.

$$\underset{x\to 0}{\lim }\frac{\tan 5x}{x}=5$$

Q6. Evaluate ${\lim }_{x\to 0}\frac{\sin x}{x^2}$.

$$\underset{x\to 0}{\lim }\frac{\sin x}{x^2}\to \mathrm{\infty }$$

Q7. Evaluate ${\lim }_{x\to 0}\frac{1-\cos 2x}{x^2}$.

$$\underset{x\to 0}{\lim }\frac{1-\cos 2x}{x^2}=2$$

Q8. Evaluate ${\lim }_{x\to 0}\frac{\sin ax}{x}$.

$$\underset{x\to 0}{\lim }\frac{\sin ax}{x}=a$$

Q9. Evaluate ${\lim }_{x\to 0}\frac{\tan bx}{x}$.

$$\underset{x\to 0}{\lim }\frac{\tan bx}{x}=b$$

Q10. Evaluate ${\lim }_{x\to \mathrm{\infty }}{(1+\frac{1}{x})}^x$.

$$\underset{x\to \mathrm{\infty }}{\lim }{(1+\frac{1}{x})}^x=e$$

FAQs (10)

1. What is a limit? Value a function approaches as input approaches a point.

2. What is ${\lim }_{x\to 0}\frac{\sin x}{x}$? Equal to 1.

3. What is ${\lim }_{x\to 0}\frac{\tan x}{x}$? Equal to 1.

4. What is ${\lim }_{x\to 0}\frac{1-\cos x}{x^2}$? Equal to $\frac{1}{2}$.

5. What is ${\lim }_{x\to \mathrm{\infty }}(1+\frac{1}{x}{)}^x$? Equal to $e$.

6. Can limit be infinite? Yes, if function grows without bound.

7. What is algebra of limits? Rules for addition, multiplication, division of limits.

8. What is standard limit of $\sin ax\mathrm{/}x$? Equal to $a$.

9. What is standard limit of $\tan bx\mathrm{/}x$? Equal to $b$.

10. Why is this exercise important? It builds foundation for derivatives and calculus.

Conclusion

Exercise 12.1 has 10 solved questions and 10 FAQs that strengthen your understanding of limits in calculus. This builds the foundation for derivatives and continuity in Class 11 Maths.

visit:www.fuzymathacademy.com

New Syllabus-Class 11 Maths Chapter 12 Limits and Derivatives – Exercise 12.1 NCERT