Evaluate each limit$.$ Do not use L$\text{'}$H$$pital$\text{'}$s rule.

$\underset{x{0}^{+}}{lim}\sqrt[\phantom{2}]{x}sin(\frac{2}{x^2})($Hint: Squeeze Theorem.$)$

$\underset{t0}{lim}\frac{tan(4t)}{tsec(4t)}$

$\underset{z0}{lim}\frac{cosz-co{s}^{2}z}{z}$

$\underset{x}{lim}\frac{2x^5+3x^4-31x}{8x^4-31x^2+12}$

$\underset{x-}{lim}\frac{8x+7x^{\frac{1}{3}}}{\sqrt[\phantom{2}]{16x^2+6}}$

Find the horizontal asymptote$($s$)$ of the function $f(t)=\frac{e^t}{1+e^{-t}}$ without using a graphing calculator.

Show using the Intermediate Value Theorem that the equation $cos(x)=x$ has a solution on the interval $0,1.($Hint: this amounts to showing $f(x)=cos(x)-x$ has a zero on $0,1.)$