Maths · Limits, continuity and differentiability
Evaluate ^{1 / x}-e}{x}=? \)
Evaluate \( : \lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e}{x}=? \)
- A. 1
- B. e/2
- C. \( -\mathrm{e} / 2 \)
- D. \( 2 / c \)
Step-by-step solution
Use the expansion: (1+x)^{1/x} = e^{\frac{1}{x}\ln(1+x)} = e^{1 - x/2 + O(x^2)} = e\left(1 - \frac{x}{2} + O(x^2)\right). Then \frac{(1+x)^{1/x} - e}{x} = \frac{-e x/2 + O(x^2)}{x} = -e/2 + O(x) \to -e/2.
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