Explain the meaning of lim x→−∞ f(x)=10.

Which of the following functions are continuous for all values in their domain? Justify your answers.

a. a(t)=altitude of a skydiver t seconds after jumping from a plane

Which of the following functions are continuous for all values in their domain? Justify your answers.

b. n(t)=number of quarters needed to park legally in a metered parking space for t minutes

Which of the following functions are continuous for all values in their domain? Justify your answers.

c. T(t)=temperature t minutes after midnight in Chicago on January 1

Which of the following functions are continuous for all values in their domain? Justify your answers.

d. p(t)=number of points scored by a basketball player after t minutes of a basketball game

Evaluate each limit and justify your answer. 

lim x→∞(2x+1x / x)^3

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)

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>

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 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. 

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

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

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

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

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 $p\left(t\right)=\frac{3500t}{t+1}$.

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

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 each limit. 

lim x→0 cos x−1 / sin^2x

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

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 continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.

$g\left(x\right)=\mid \frac{x+4}{x^2-4}\mid$

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

lim x→3 x − 3 /|x − 3|

Determine the following limits.​

lim x→1 x^3 − 7x^2 + 12x / 4 − x

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4

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

lim x→1^+ 1 /1 − x=−∞

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

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

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

Determine the following limits.​

lim u→0^+ u − 1 / sin u

Determine the following limits. 

lim x→∞ sin x / e^x