91Ó°ÊÓ

(a) find the intervals on which \(f\) is increasing or decreasing, and (b) find the relative maxima and relative minima of \(\vec{f}\). $$ f(x)=\frac{2 x-3}{x^{2}-4} $$

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 0} \frac{\sin x-x}{e^{x}-e^{-x}-2 x} $$

In Exercises \(25-40\), find the critical number \((s)\), if any, of the function. $$ f(x)=2 x+3 $$

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ f(x)=\sin 2 x, \quad 0 \leq x \leq \pi $$

Estimate the value of the radical accurate to four decimal places by using three iterations of Newton's method to solve the equation \(f(x)=0\) with initial estimate \(x_{0}\). \(\sqrt[3]{7} ; \quad f(x)=x^{3}-7 ; x_{0}=2\)

In Exercises \(25-40\), find the critical number \((s)\), if any, of the function. $$ g(x)=4-3 x $$

Find the limit. $$ \lim _{x \rightarrow-\infty} \frac{x^{4}+1}{x^{3}+1} $$

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ g(x)=2 \sin x+\sin 2 x, \quad 0 \leq x \leq 2 \pi $$

(a) find the intervals on which \(f\) is increasing or decreasing, and (b) find the relative maxima and relative minima of \(\vec{f}\). $$ f(x)=\frac{x^{2}-3 x+2}{x^{2}+2 x+1} $$

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ g(x)=\cos ^{2} x, \quad 0 \leq x \leq 2 \pi $$