Chapter 6

Find the limits of the following: \(\lim _{x \rightarrow 0}(x-5) \cos x\)

Find the limits of the following: If \(b \neq 0\), evaluate \(\lim _{x \rightarrow b} \frac{x^{3}-b^{3}}{x^{6}-b^{6}}\).

Find the limits of the following: \(\lim _{x \rightarrow-\infty} \frac{x^{2}+2 x-3}{x^{3}+2 x^{2}}\)

Find the limits of the following: \(\lim _{x \rightarrow \infty} \frac{3 x^{2}}{5 x+8}\)

Find the limits of the following: \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}-4}}\)

Find the limits of the following: . If \(f(x)=\left\\{\begin{array}{cl}e^{x} & \text { for } \quad 0 \leq x<1 \\\ x^{2} e^{x} & \text { for } \quad 1 \leq x \leq 5\end{array}\right.\), find \(\lim _{x \rightarrow 1} f(x)\)

Find the limits of the following: \(\lim _{x \rightarrow \infty} \frac{e^{x}}{1-x^{3}}\)

Find the horizontal and vertical asymptotes of the graph of the function \(f(x)=\frac{1}{x^{2}+x-2} .\)

Find all \(x\) -values where the function \(f(x)=\frac{x+1}{x^{2}+4 x-12}\) is discontinuous.

For what value of \(k\) is the function \(g(x)=\left\\{\begin{array}{ll}x^{2}+5, & x \leq 3 \\ 2 x-k, & x>3\end{array}\right.\) continuous at \(x=3\) ?

Determine if \(f(x)=\left\\{\begin{array}{ll}\frac{x^{2}+5 x-14}{x-2}, & \text { if } x \neq 2 \\ 12, & \text { if } x=2\end{array}\right.\) continuous at \(x=2\). Explain why or why not.

A function \(f\) is continuous on \([-2,2],\) and some of the values of \(f\) are shown below: $$\begin{array}{|c|c|c|c|}\hline x & -2 & 0 & 2 \\\\\hline f(x) & 3 & b & 4 \\\\\hline\end{array}$$ If \(f\) has only one root, \(r,\) on the closed interval \([-2,2],\) and \(r \neq 0,\) then a possible value of \(b\) is (A) -2 (B) -1 (C) 0 (D) 1

Evaluate \(\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}\)

Write an equation of the line passing through the point (2,-4) and perpendicular to the line \(3 x-2 y=6\)

Evaluate \(\lim _{x \rightarrow 0} \frac{|3 x-4|}{x-2}\)

Find \(\lim _{x \rightarrow 0} \frac{\tan x}{x}\)

Find the horizontal and vertical asymptotes of \(f(x)=\frac{x}{\sqrt{x^{2}+4}}\).