JAMIA 2021 — Mathematics PYQ

1. Definition of Continuity

For $f(x)$ to be continuous at $x=0$:

2. Evaluate the Limit

Substitute the given function into the limit expression:

3. Use the Squeeze Theorem (Sandwich Theorem)

We know that the sine function oscillates between $-1$ and $1$ for any real input:

Multiply the entire inequality by $x^2$ (since $x^2\ge 0$, the inequality signs remain the same):

4. Apply the limit as $x\to 0$

Calculate the limits of the boundary functions:

Since both the lower and upper bounds approach $0$, the middle function must also approach $0$:

5. Final Value

Therefore, for the function to be continuous: