The value of the following limit is _____________. _x 0^+ x1 - e^2x

GATE CSE 2022 Q34: Limit of √x / (1 − e^(2√x)) as x → 0+

The value of the following limit is _____________.

$\lim_{x \to 0^+} \frac{\sqrt{x}}{1 - e^{2\sqrt{x}}}$

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Answer: -0.5 (official GATE 2022 final answer key)

Concept used: A $\frac{0}{0}$ form: substitute $t = \sqrt{x}$ and use $e^{u} \approx 1 + u$ near 0 (or L'Hôpital).

First step: With $t = \sqrt{x} \to 0^+$ , the limit is $\lim_{t \to 0^+} \frac{t}{1 - e^{2t}}$ .

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Source: official GATE 2022 CS question paper and final answer key, organised by IIT Kharagpur. Question paper · Answer key