Determine the following limits.


b. $\underset{x\to 3}{\lim }\frac{x-3}{x^4-9x^2}$

The hyperbolic cosine function, denoted $\(\cosh\]\left\)(x\(\right\))$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}$.

b. Evaluate $\cosh \left(0\right)$. Use symmetry and part (a) to sketch a plausible graph for $y=\cosh \left(x\right)$.

Assume you invest \(250 at the end of each year for 10 years at an annual interest rate of $r$. The amount of money in your account after 10 years is given by $A\left(r\right)=\frac{250({(1+r)}^{10}-1)}{r}$. Assume your goal is to have \)3500 in your account after 10 years.

b. Use a calculator to estimate the interest rate required to reach your financial goal.

Let $g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)$

b. Determine the value of $a$ for which $g$ is continuous from the right at $1$. 

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of f (if any).

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

b. lim x→−2 f(x)

Use the graph of $g$ in the figure to find the following values or state that they do not exist. <IMAGE>

${\(\displaystyle\]\lim\)_{x\(\to\)0}g\(\left\)(x\(\right\))}$