WBJEE · Maths · Limits
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
- A does not exist
- B exists and equals 1
- C exists and equals 0
- D exists and equals \(\frac{1}{2}\)
Answer & Solution
Correct Answer
(C) exists and equals 0
Step-by-step Solution
Detailed explanation
\(I=\lim _{x \rightarrow 0} \sin \left(\frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}\right)=\lim _{x \rightarrow 0} \frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}=\lim _{x \rightarrow 0} \frac{e^{x}-1-0-x}{2 x}=\lim _{x \rightarrow 0} \frac{e^{x}-1}{2}=0\)
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