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Question: Answered & Verified by Expert
If $\mathrm{I}=\lim _{\mathrm{x} \rightarrow 0} \sin \left(\frac{\mathrm{e}^{\mathrm{x}}-\mathrm{x}-1-\frac{\mathrm{x}^{2}}{2}}{\mathrm{x}^{2}}\right)$, then limit
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Options:
  • A does not exist
  • B exists and equals 1
  • C exists and equals 0
  • D exists and equals $\frac{1}{2}$
Solution:
1919 Upvotes Verified Answer
The correct answer is: exists and equals 0
$I=\lim _{x \rightarrow 0} \sin \left(\frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}\right)=\lim _{x \rightarrow 0} \frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}=\lim _{x \rightarrow 0} \frac{e^{x}-1-0-x}{2 x}=\lim _{x \rightarrow 0} \frac{e^{x}-1}{2}=0$

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