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).

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

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

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.

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

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

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

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)

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

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>

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

Evaluate each limit. 

lim x→0 cos x−1 / sin^2x

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

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^2−25 / x−5; a=5

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→1 (8x+5)=13

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

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).

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

Evaluate each limit. 

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

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

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4

Determine the following limits.

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

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

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

lim x→1^+ 1 /1 − x=−∞

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. 

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

Determine the following limits.​

lim u→0^+ u − 1 / sin u

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).

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

Determine the following limits. 

lim x→∞ sin x / e^x

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

lim x→3 −5x / √4x − 3

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

f(x)=1/ √x sec x

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

x^5+7x+5=0; (−1,0)