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 culture of tumor cells is given by $p\left(t\right)=\frac{3500t}{t+1}$.

Evaluate $\underset{x\to \infty }{\lim }f\left(x\right)$ and$\underset{x\to -\infty }{\lim }f\left(x\right)$.

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

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 $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

Determine the following limits.

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

Determine the following limits.

$\underset{x\to 0^{-}}{\lim }\csc \left(x\right)$

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

Determine the following limits.​

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

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

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 

f(x)=2x^2+3x+1 / x^2+5x; a=−5

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

f(x)=x^5+6x+17 / x^2−9

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).

$f\left(x\right)=\frac{1}{2x^4-\sqrt{4x^8-9x^4}}$

Determine the following limits.​

lim w→∞ (ln w²) / (ln w³ + 1)

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→0 x^2=0 (Hint: Use the identity √x2=|x|.)

Evaluate lim x→0 x + 1/ 1 −cos x.

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→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)

Evaluate each limit. 

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

Which one of the following intervals is not symmetric about x=5?

a.(1, 9)

b.(4, 6)

c.(3, 8)

d.(4.5, 5.5)

Determine the following limits.

$\underset{z\to 4}{\lim }\frac{z-5}{{(z^2-10z+24)}^2}$

Determine the following limits. 

lim x→∞ sin x / e^x

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).

$f\left(x\right)=4x(3x-\sqrt{9x^2+1})$

Suppose ${\displaystyle\lim_{x\to a}}\text{ }f\left(x\right)=L$. Prove that $\underset{x\to a}{\lim }\text{ (}c\left(f\left(x\right)\right)=cL$, where $c$ is a constant.

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).

$f\left(x\right)=\frac{\sqrt[3]{x^6+8}}{4x^2+\sqrt{3x^4+1}}$

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

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

Evaluate each limit. 

$\underset{x\to \pi }{\lim }\frac{{\cos }^2\left(x\right)+3\cos \left(x\right)+2}{{\cos }^{}\left(x\right)+1}$

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

$\underset{x\to 0}{\lim }\frac{1-\cos \left(x\right)}{{\cos }^2\left(x\right)-3\cos \left(x\right)+2}$

Evaluate each limit and justify your answer. 

lim x→0 (x^8−3x^6−1)^40

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

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,…

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

x^5+7x+5=0; (−1,0)