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 lim x→1 (x^3+3x^2−3x+1).

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

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

Determine the following limits.​

lim x→∞ (2x − 3) / (4x + 10)

Determine the following limits.​

lim x→∞ (3 tan^-1 x + 2)

Determine the following limits.​

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

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

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

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

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

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.

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 $x=1$.

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].

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

x=cos x; (0,π/2)

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.

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocit​​y at t=1. <IMAGE>

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>

Tangent lines​​ with zero slope

a. Graph the function f(x)=x^2−4x+3.

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

a. lim x→−2^+ f(x)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

b. lim x→−2 f(x)

Determine the following limits at infinity.

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

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

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

Calculate $g\left(x\right)$ for each value of $x$ in the following table.

Determine the following limits.​

lim x→1 x^3 − 7x^2 + 12x / 4 − x

Determine the following limits.​

lim x→3 1/ x − 3(1 /√x + 1 − 1/2)

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.

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