Second Year Math Functions and Limits

4. Express each limit in terms of e :\[\text { (x) } \operatorname{Lim}_{x \rightarrow 0} \frac{e^{1 / x}-1}{e^{1 / x}+1} x<0\]

$Example 1:Graph the circle x^{2}+y^{2}=4$

$4. Express each limit in terms of e :(ix) \operatorname{Lim}_{x \rightarrow \infty}\left(\frac{x}{1+x}\right)^{x}$

$1. Evaluate each limit by using theorems of limits:(v) \operatorname{Lim}_{x \rightarrow 2}\left(\sqrt{x^{3}+1}-\sqrt{x^{2}+5}\right)$

Example 1: Determine whether \operatorname{Lim}_{x \rightarrow 2} f(x) and \operatorname{Lim}_{x \rightarrow 4} f(x) exist when\[f(x)=\left\{\begin{array}{rcc}2 x+1 & \text { if } & 0 \leq x \leq 2 \\7-x & \text { if } & 2 \leq x \leq 4 \\x & \text { if } & 4 \leq x \leq 6\end{array}\right.\]

$Example 7:Evaluate: \operatorname{Lim}_{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}$