$\lim _{\mathrm{x} \rightarrow 0} \frac{\mathrm{x}\left(\mathrm{e}^{\left(\sqrt{1+\mathrm{x}^{2}+\mathrm{x}^{4}-1}\right) / \mathrm{x}}-1\right)}{\sqrt{1+\mathrm{x}^{2}+\mathrm{x}^{4}}-1}$

(C) is equal to $\sqrt{\mathrm{e}}$

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Limits

lim_{x \to 0} \frac{x ( e ^{(1 + x^{2} + x^{4} - 1) / x} - 1 )}{1 + x ^{2} + x ^{4} - 1}

Answer: (B)

Let

t = \frac{1 + x ^{2} + x ^{4} - 1}{x}

, which tends to

0

x \to 0

. Pairing the leading

x

with the denominator

1 + x^{2} + x^{4} - 1 = x t

, the expression is exactly

\frac{e ^{t} - 1}{t}

. Since

\frac{e ^{t} - 1}{t} \to 1

t \to 0

, the limit is

1

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Single correct option

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