The value of limx→0 ex−esin⁡x2(x−sin⁡x), is

A
$- \frac{1}{2}$
B
$\frac{1}{2}$
C
1
D
$\frac{3}{2}$

Solution:

We have,

$\begin{array}{l} lim_{x \to 0} \frac{e^{x} - e^{\sin x}}{2 (x - \sin x)} \\ = \frac{1}{2} lim_{x \to 0} e^{\sin x} (\frac{e^{x - \sin x} - 1}{x - \sin x}) = \frac{1}{2} \times e^{0} \times 1 = \frac{1}{2} \end{array}$

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