Question

Assertion :$\underset{x\to -1}{lim}\frac{\sqrt[3]{37-x}-2}{(x+1)}=-\frac{1}{12}$ Reason: $\underset{x\to a}{lim}=\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$L=\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}$
Applying L'Hospital rule
$L=\underset{x\to a}{lim}{nx}^{n-1}=n{a}^{n-1}$
Now $\underset{x\to -1}{lim}\frac{\sqrt[3]{37-x}-2}{x+1}$
Applying L'Hospital rule
$=\underset{x\to -1}{lim}\frac{-{(7-x)}^{-\frac{2}{3}}}{3}=-\frac{1}{3{\left(8\right)}^{\frac{2}{3}}}=-\frac{1}{12}$

Hence both reason and assertion are true, reason is not the corrrect explaination of the assertion