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

f(x)=1 / x^2−4

Determine the following limits.

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

Evaluate each limit. 

$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(x\right)-1}{\sqrt{\sin \left(x\right)}-1}$

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

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

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

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

Determine the following limits. 

lim x→−∞ (3x⁷ + x²) 

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

f(x)=1/ √x sec x

If a function f represents a system that varies in time, the existence of lim ${\displaystyle\lim_{t\rightarrow\infty}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a colony of squirrels is given by $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

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

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 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

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$, determine ${\displaystyle\lim_{x\to a^{+}}}f\left(x\right)$, ${\displaystyle\lim_{x\to a^{-}}}f\left(x\right)$, and ${\displaystyle\lim_{x\to a}}f\left(x\right)$.

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

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→3 x^3=27

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)

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

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

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(z)=(z−1)^3/4

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{x-2}{\ln \mid x-2\mid }$; $a=2$

Determine the following limits.​

lim x→0^− 2 / tan x

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→∞ (2x − 3) / (4x + 10)

Determine the following limits.​

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

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.

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

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

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

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

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

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

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.

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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 $x=1$.