Question

Assertion :If $\underset{x\to 0}{lim}\left\{f\right(x)+\frac{\sin x}{x}\}$ does not exist, then ${lim}_{x\to 0}f\left(x\right)$ does not exist. Reason: $\underset{x\to 0}{lim}\frac{\sin x}{x}$ exists and has value 1.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

If ${lim}_{x\to 0}f\left(x\right)$ exists, then $\underset{x\to 0}{lim}\left(f\right(x)+\frac{sinx}{x})$ always exists
as $\underset{x\to }{lim}\frac{sinx}{x}$ exists
finitely.
Hence, ${lim}_{x\to 0}f\left(x\right)$ must not exist.