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## An example explains the impotance of assumption in L’Hospital’s Rule

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 30 Exercises 30.6 Solution: Part aNote that $\cos x\sin x\ge -1$ and $\sin x\ge -1$, we have$$\lim_{x\to\infty}f(x)\ge \lim_{x\to\infty}(x-1)=\infty.$$ Hence…

## Solution to Mathematics for Machine Learning Exercise 5.3

Compute the derivative $f’(x)$ of the function $$f(x)=\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right).$$ Solution: Clearly, we have $$\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)’=-\frac{1}{2\sigma^2}2(x-\mu)=\frac{-(x-\mu)}{\sigma^2}.$$Therefore, by Chain rule (5.32), we have \begin{align*}f’(x)=&\ \exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\cdot \left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)’\\=&\ \frac{-(x-\mu)}{\sigma^2}\cdot\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\end{align*}