Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure.​​ <IMAGE>

Determine the following limits.

$\underset{\theta \to 0^{-}}{\lim }\frac{\sin \theta }{{\cos }^2\theta -1}$

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(x)=(2x−3)^2/3

Determine the following limits.​

lim x→0^− 2 / tan x

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={2x if x<1

x^2+3x if x≥1; a=1

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

Evaluate lim x→0 x + 1/ 1 −cos x.

Determine the following limits. 

lim x→−∞ (3x⁷ + x²) 

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 

{2,3/4,4/9,5/16,…}, which is defined by f(n) = (n+1) / n^2, for n=1,2,3,…

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

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

Evaluate f(3) if lim x→3^− f(x)=5,lim x→3^+ f(x)=6, and f is right-continuous at x=3.

Determine the following limits.

$\underset{x\to 0}{\lim }\frac{x^3-5x^2}{x^2}$

Determine the following limits. 

lim θ→∞ cos θ / θ²

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

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

g(θ)=tan πθ/10

Determine the following limits at infinity.

lim t→∞ (−12t^−5)

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(z)=(z−1)^3/4

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>

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

$\underset{x\to 0}{\lim }(\frac{1}{x^2}+1)=\infty$

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

$g\left(x\right)=\cos \left(e^x\right)$

b. Estimate a solution to the equation in the given interval using a root finder.

x=cos x; (0,π/2)

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

Determine the following limits.​

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

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

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

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

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

$f\left(x\right)=1-e^{-2x}$

Determine the following limits.​

lim x→∞ (2x − 3) / (4x + 10)

Use the graph of $f$ in the figure to determine the values of $x$ in the interval $(-3,5)$ at which f fails to be continuous. Justify yo​​ur answers using the continuity checklist.

<IMAGE>

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

Determine values of the constants $a$ and $b$ , if possible, for which $g$ is continuous at $x=1$.