Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

h(x)=e^x(x+1)^3

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to \infty }{\lim }x\cos \left(x\right)$

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4

Suppose $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$. Prove that $\underset{x\to a}{\lim }(f\left(x\right)-g\left(x\right))=L-M$.

Evaluate each limit. 

$\underset{x\to 3}{\lim }\sqrt{x^2+7}$

Determine the interval(s) on which the following functions are continuous. 

f(x)=x^5+6x+17 / x^2−9

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

The following table gives the position $s\left(t\right)$ of an object moving along a line at time $t$. Determine the average velocities over the time intervals $\left\lbrack1,1.01\right\rbrack$, $\left\lbrack1,1.001\right\rbrack$, and $\left\lbrack1,1.0001^{}\right.]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>4

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.

g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

Describe the end behavior of g(x) = e^-2x.

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

Find an interval containing a solution to the equation $2x=\cos \left(x\right)$. Use a graphing utility to approximate the solution.

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

$f(-1)=-2$, $f\left(1\right)=2$, $f\left(0\right)=0$, $\underset{x\to \infty }{\lim }f\left(x\right)=1$, $\underset{x\to -\infty }{\lim }f\left(x\right)=-1$

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=\sin x$

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to 4}{\lim }\frac{1}{{(x-4)}^2}=\infty$

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2+x−2 / x−1; a=1

Evaluate ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and${\displaystyle\lim_{x\to-\infty}{f(x)}}$.

$f\left(x\right)=\frac{6e^x+20}{3e^x+4}$

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=x^2−3x+2 / x^10−x^9

Determine the interval(s) on which the following functions are continuous. 

p(x)=4x^5−3x^2+1

Determine the following limits.​

lim x→π/2 1/√sin x − 1 / x + π/2

Evaluate lim x→1 (x^3+3x^2−3x+1).

Determine the following limits.

$\underset{x\to 1^{+}}{\lim }\frac{x^2-5x+6}{x-1}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{3x^3-7}{x^4+5x^2}$

Use the graph of f(x) = x / (x² − 2x − 3)² to determine lim x→−1 f(x) and lim x→3 f(x).