Question

高数的求极限问题

第七题

Answer 1

LaTex代码:

\begin{aligned} 

&\quad\; \lim_{x\to+\infty} \left(e^{\sqrt{1-\sin \frac{1}{x}}}-e \right) x\\

&=\lim_{x\to+\infty} e^{\xi}\left(\sqrt{1-\sin \frac{1}{x}} - 1 \right ) x && \left(\text{拉格朗日中值定理}, \xi \in \left(\sqrt{1-\sin \frac{1}{x}}, 1 \right )\right)\\

&=\lim_{x\to+\infty} e\cdot\left(-\frac{1}{2x}\right)\cdot x = -\frac{e}{2} && \left(\sqrt{1-\sin \frac{1}{x}} - 1 \sim \frac{1}{2}\left(-\sin\frac{1}{x}\right)\sim -\frac{1}{2x}\right)

\end{aligned}