Class 12 Maths Chapter 5 Continuity and Differentiability – Exercise 5.1 NCERT Solutions

Introduction

Exercise 5.1 introduces the concepts of continuity of functions. Students learn how to check continuity at a point, over an interval, and apply limits to prove continuity. This exercise builds the foundation for differentiability and calculus.

Formulas Used

1. Continuity at a Point: A function $f(x)$ is continuous at $x=a$ if:

$$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a)$$

2. Limit Laws:

$$\underset{x\to a}{\lim }[f(x)+g(x)]=\underset{x\to a}{\lim }f(x)+\underset{x\to a}{\lim }g(x)$$

$$\underset{x\to a}{\lim }[f(x)\cdot g(x)]=\underset{x\to a}{\lim }f(x)\cdot \underset{x\to a}{\lim }g(x)$$

3. Polynomial and Rational Functions:

   • Polynomials are continuous everywhere.

   • Rational functions are continuous wherever denominator ≠ 0.

Students Frequently Make Mistakes

• Forgetting to check both left‑hand and right‑hand limits.

• Confusing continuity with differentiability.

• Ignoring domain restrictions in rational functions.

• Errors in evaluating limits.

• Skipping substitution step for polynomials.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Check continuity of $f(x)=x^2$ at $x=2$.

$$\underset{x\to 2^{-}}{\lim }f(x)=4,\underset{x\to 2^{+}}{\lim }f(x)=4,f(2)=4$$

Hence continuous.

Q2. Check continuity of $f(x)=\frac{1}{x}$ at $x=0$. Denominator zero → not defined → discontinuous at $x=0$.

Q3. Check continuity of $f(x)=\mathrm{\mid }x\mathrm{\mid }$ at $x=0$.

$$\underset{x\to 0^{-}}{\lim }\mathrm{\mid }x\mathrm{\mid }=0,\underset{x\to 0^{+}}{\lim }\mathrm{\mid }x\mathrm{\mid }=0,f(0)=0$$

Hence continuous.

Q4. Check continuity of $f(x)=\sin x$ at $x=\pi \mathrm{/}2$.

$$\underset{x\to \pi \mathrm{/}2}{\lim }\sin x=1,f(\pi \mathrm{/}2)=1$$

Continuous.

Q5. Check continuity of $f(x)=\cos x$ at $x=0$.

$$\underset{x\to 0}{\lim }\cos x=1,f(0)=1$$

Continuous.

Q6. Check continuity of $f(x)=\tan x$ at $x=\pi \mathrm{/}2$. Not defined at $x=\pi \mathrm{/}2$. Discontinuous.

Q7. Check continuity of $f(x)=x^3$ at $x=1$.

$$\underset{x\to 1}{\lim }{x}^3=1,f(1)=1$$

Continuous.

Q8. Check continuity of $f(x)=\frac{x^2-1}{x-1}$ at $x=1$. Simplify: $\frac{(x-1)(x+1)}{x-1}=x+1,x\mathrm{\ne }1$.

$$\underset{x\to 1}{\lim }f(x)=2,f(1)\text{ not defined}$$

Discontinuous at $x=1$.

Q9. Check continuity of $f(x)=\sqrt{x}$ at $x=4$.

$$\underset{x\to 4}{\lim }\sqrt{x}=2,f(4)=2$$

Continuous.

Q10. Check continuity of $f(x)=e^x$ at $x=0$.

$$\underset{x\to 0}{\lim }{e}^x=1,f(0)=1$$

Continuous.

FAQs (10)

FAQ1. What is continuity at a point? Equality of left limit, right limit, and function value.

FAQ2. Are polynomials continuous everywhere? Yes.

FAQ3. Are rational functions continuous everywhere? Yes, except where denominator = 0.

FAQ4. Is $\mathrm{\mid }x\mathrm{\mid }$ continuous at 0? Yes.

FAQ5. Is $\tan x$ continuous at $\pi \mathrm{/}2$? No, undefined.

FAQ6. Is exponential function continuous everywhere? Yes.

FAQ7. Is logarithmic function continuous everywhere? Yes, for domain $x>0$.

FAQ8. Is trigonometric function continuous everywhere? Yes, except where undefined.

FAQ9. What is difference between continuity and differentiability? Differentiability implies continuity, but not vice versa.

FAQ10. Why is Exercise 5.1 important? It builds foundation for differentiability and calculus.

Conclusion

Exercise 5.1 has 10 solved questions and 10 FAQs that strengthen your understanding of continuity of functions. This builds the foundation for differentiability and calculus in Class 12 Maths.

visit:www.fuzymathacademy.comwww.fuzymathacademy.com