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|.)

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→2 (x^2+3x)=10

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 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

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 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→5 1/x^2=1/25

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^- 1 / 1 − x=∞

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L

d. If |x−a|<δ, then a−δ<x<a+δ.

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

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.

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

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

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

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

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

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

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

$\underset{x\to 0}{\lim }(\frac{1}{x^4}-\sin \left(x\right))=\infty$

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.

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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

f(x) = (x² − 9)/(x(x−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)

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

x^5+7x+5=0; (−1,0)

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

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. 

{0,1/2,2/3,3/4,…}, which is defined by f(n) = (n−1) / n, for n=1,2,3,…

Determine the following limits at infinity.

lim x→∞ (3+10/x^2)

Complete the following sentences in terms of a limit.

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

Complete the following sentences in terms of a limit.

b. A function is continuous from the right at a if _____ .

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,…

What does it mean for a function to be continuous on an interval?

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\left(2\right)$

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\left(4\right)$

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

${\displaystyle\lim_{x\to4}h\left(x\right)}$