MCQ
- Ae
- B$\frac{-2}{ e }$
- C0
- D$e-e^2$
✓
Answer
$\operatorname{Lim}_{x \rightarrow 0} \frac{e-e^{\frac{1}{2 x} \ln (1+2 x)}}{x}$
$=\operatorname{Lim}_{x \rightarrow 0}(-e) \frac{\left(e^{\frac{\ln (1+2 x)}{2 x}-1}-1\right)}{x}$
$=\operatorname{Lim}_{x \rightarrow 0}(-e) \frac{\ln (1+2 x)-2 x}{2 x^2}$
$=(-e) \times(-1) \frac{4}{2 \times 2}=e$
$=\operatorname{Lim}_{x \rightarrow 0}(-e) \frac{\left(e^{\frac{\ln (1+2 x)}{2 x}-1}-1\right)}{x}$
$=\operatorname{Lim}_{x \rightarrow 0}(-e) \frac{\ln (1+2 x)-2 x}{2 x^2}$
$=(-e) \times(-1) \frac{4}{2 \times 2}=e$
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