lim x → 0 ⁡ e x 2 - cos ⁡ x sin 2 ⁡ x is equal to

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Question: Answered & Verified by Expert

\lim_{x \to 0} \frac{e^{x^{2}} - \cos x}{\sin^{2} x}

is equal to

MathematicsLimitsJEE MainJEE Main 2015 (10 Apr Online)

Options:

A $2$
B $\frac{3}{2}$
C $\frac{5}{4}$
D $3$

Solution:

1381 Upvotes Verified Answer

The correct answer is:

\frac{3}{2}

\lim_{x \to 0} \frac{e^{x^{2}} - \cos x}{\sin^{2} x} = \lim_{x \to 0} \frac{(e^{x^{2}} - 1) + (1 - \cos x)}{x^{2} \cdot (\frac{\sin^{2} x}{x^{2}})}

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

= 1 + \frac{1}{2} = \frac{3}{2}

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