limx → ∞ x1/x is A) ∞ B) 0 C) 1 D) Not defined

Official answer: C) 1. Explanation: Let the limit be: y = limx → ∞ x1/x Taking the natural logarithm on both sides: ln y = limx → ∞ ln (x1/x ) = limx → ∞ (ln x)/(x) As x → ∞, this is an indeterminate form of type (∞)/(∞). Applying L'Hôpital's Rule: limx → ∞ (d)/(dx)(ln x)(d)/(dx)(x) = limx → ∞ (1)/(x)1 = 0 Thus: ln y = 0 y = e0 = 1 Therefore, limx → ∞ x1/x = 1, which corresponds to option (C).

Calculus

GATE CSE 2015 Set 1 · Q4

MCQ1 markEasy

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$\lim_{x \to \infty} x^{1/x}$ is

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Official answer: C) 1

Calculus

Let the limit be:

y = \lim_{x \to \infty} x^{1/x}

Taking the natural logarithm on both sides:

\ln y = \lim_{x \to \infty} \ln\left(x^{1/x}\right) = \lim_{x \to \infty} \frac{\ln x}{x}

As $x \to \infty$ , this is an indeterminate form of type $\frac{\infty}{\infty}$ . Applying L'Hôpital's Rule:

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

Thus:

\ln y = 0 \implies y = e^0 = 1

Therefore, $\lim_{x \to \infty} x^{1/x} = 1$ , which corresponds to option (C).

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This Calculus previous year question is Q4 of the GATE CSE 2015 Set 1 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

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Calculus in GATE CSE

Calculus has 40 GATE CSE PYQs across 25 papers from 2011 to 2026. 14 of them are 1-mark MCQs like this one, and GATE CSE 2015 Set 1 has 4.

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