Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$h\left(x\right)=\frac{2x}{x^3-25x}$

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

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

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 $\cosh \left(x\right)=\frac{e^x+e^{-x}}{2}$.

a. Determine its end behavior by analyzing $\underset{x\to \infty }{\lim }\cosh \left(x\right)$ and $\underset{x\to -\infty }{\lim }\cosh \left(x\right)$.

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.

f(x)=x^1/3 <IMAGE>

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

Calculate $g\left(x\right)$ for each value of $x$ in the following table.

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

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

Determine the following limits.

a. $\underset{x\to 2^{+}}{\lim }\frac{1}{\sqrt{x(x-2)}}$

Complete the following sentences in terms of a limit.

a. A function is continuous from the left at a if _____.

Tangent lines​​ with zero slope

a. Graph the function f(x)=x^2−4x+3.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

a. Graph the position function, for 0≤t≤9.

Determine whether the following statements are true and give an explanation or counterexample.

a. The graph of a function can never cross one of its horizontal asymptotes.

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^− h(x)

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.​ 

a. s(t)=−16t^2+80t+60 at t=3

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.

a. lim x→−2^+ f(x)

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x² − 9)/(x(x−3))

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^− f(x)

Determine the following limits.

a. $\underset{x\to 1^{+}}{\lim }\frac{x-3}{\sqrt{x^2-5x+4}}$

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground? 

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

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.

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^+ h(x)

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 $y=\cosh \left(x\right)$.

Determine the following limits.

b. $\underset{x\to 2}{\lim }\frac{x-2}{x^5-4x^3}$

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (x^4−1)/(x^2−1)

Complete the following steps for the given functions. 

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

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Complete the following steps for the given functions. 

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

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

Complete the following steps for the given functions. 

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

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