NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

Which of the following is not true?

Choose the correct answer:

Explanation

Evaluating Option A:

\lim_{x \to \infty} \frac{x}{e^x}

This is an indeterminate form of $\frac{\infty}{\infty}$ . Applying L'Hôpital's Rule:

\lim_{x \to \infty} \frac{\frac{d}{dx}(x)}{\frac{d}{dx}(e^x)} = \lim_{x \to \infty} \frac{1}{e^x} = 0

So, Option A is true.

Evaluating Option B:

\lim_{x \to 0^+} \frac{1}{xe^{1/x}}

As $x \to 0^+$ , let $t = 1/x$ . As $x \to 0^+$ , $t \to \infty$ . The expression becomes:

\lim_{t \to \infty} \frac{t}{e^t}

As proven in Option A, this limit is $0$ .

So, Option B is true.

Evaluating Option C:

\lim_{x \to 0^+} \frac{\sin x}{1+2x}

Substitute $x = 0$ directly:

\frac{\sin(0)}{1+2(0)} = \frac{0}{1} = 0

So, Option C is true.

Evaluating Option D:

\lim_{x \to 0^+} \frac{\cos x}{1+2x}

Substitute $x = 0$ directly:

\frac{\cos(0)}{1+2(0)} = \frac{1}{1+0} = 1

The statement says the limit is $0$ , but the calculated limit is $1$ .

So, Option D is not true.

Correct Answer:

D. $\lim_{x \to 0^+} \frac{\cos x}{1+2x} = 0$