Expert Answer to "Evaluate:limx→0ex-esinxx-sinx A)-1; B)0; C)1; D)None of these"

Linda Norman

Answered question

2022-12-25

Evaluate:

\lim x \to 0 \frac{e^{x} - e^{\sin x}}{x - \sin x}

- 1

;
B)

0

;
C)

1

;
D)None of these

Ioriannis4v

Beginner2022-12-26Added 11 answers

The correct answer is C

1

Explanation for the correct answer:
Applying the limits and reducing the equation to its simplest form:

\Rightarrow \lim x \to 0 (\frac{e^{x} - e^{\sin x}}{x - \sin x}) \Rightarrow \lim x \to 0 (\frac{e^{x} - e^{\sin x} \cos x}{1 - \cos x}) (Differentiating) \Rightarrow \lim x \to 0 (\frac{e^{x} - (e^{\sin x} (- \sin x) + e^{\sin x} \cos x \cdot \cos x)}{0 - (- \sin x)}) (Differentiating) \Rightarrow \lim x \to 0 (\frac{e^{x} + e^{\sin x} . \sin x - \cos^{2} x \cdot x e^{\sin x}}{\sin x}) \Rightarrow \lim x \to 0 (\frac{e^{x} + e^{\sin x} \cos x + \sin x \cdot e^{\sin x} - \cos^{2} x \cdot e^{\sin x} \cos^{x - 2 \cos x (- \sin x) e^{\sin x}}}{\cos x}) (Differentiating)

Applying the limits

\Rightarrow \frac{e^{0} + e^{0} \times 1 + 0 \cdot e^{0} \cdot 1 - 1^{2} \cdot e^{0} \cdot 1 - 2 \cdot 1 (- 0) e^{0}}{1} \Rightarrow 1 + 1 - 1 \Rightarrow 1

Thus,

\lim x \to 0 \frac{e^{x} - e^{\sin x}}{x - \sin x} = 1

Option (C) is, thus, the right response.

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