91Ó°ÊÓ

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0^{+}} \frac{e^{x}}{x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=e^{-t / 2}, t \geq 0 $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow(\pi / 2)^{-}}(\tan x+\sec x) $$

Suppose that \(f\) is differentiable for all \(x \in \mathbf{R}\) with \(f(2)=3\) and \(f^{\prime}(x)=0\) for all \(x \in \mathbf{R}\). Find \(f(x)\).

Suppose that \(f(x)=e^{-|x|}, x \in[-2,2]\). (a) Show that \(f(-2)=f(2)\). (b) Compute \(f^{\prime}(x)\), where defined. (c) Show that there is no number \(c \in(-2,2)\) such that \(f^{\prime}(c)=0\). (d) Explain why your results in (a) and (c) do not contradict Rolle's theorem. (e) Use a graphing calculator to sketch the graph of \(f(x)\).

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=1-e^{-3 t}, t \geq 0 $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow(\pi / 2)^{-}} \frac{\tan x}{1+\sec x} $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 1} \frac{x^{2}-1}{x+1} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d s}=\sin (\pi s), 0 \leq s \leq 1 $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d s}=\cos (2 \pi s), 0 \leq s \leq 1 $$