1. Limits and Continuity

Determine the following limits at infinity.

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

Suppose a contour map is given for a function $f$, and you are asked to estimate the partial derivatives $f_x(2,1)$ and $f_y(2,1)$ at the point $(2,1)$. Which of the following best describes how you would use the contour map to estimate these partial derivatives?

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos \left(4x\right)}{1+4x-e^{4x}}$.

Use a series expansion to evaluate the limit: $\underset{x\to 0}{lim}\frac{(1+x)-1-\frac{1}{2}{x}^2}{x^2}$. What is the value of this limit?

Given a curve $C$ and a contour map of a function $f$ whose gradient is continuous, which of the following statements is true about the direction of the gradient vector of $f$ at a point on $C$?

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

b. How does the graph behave as x → ±∞?

Give reasons for your answers.

y = (3/2)(x / (x − 1))²/³

6) Use the Intermediate Value Theorem to show that the equation $x^2-6x-3=0$ has a solution on the interval $[0,7]$. Which of the following justifies the existence of a solution?

The graph of $y=f\left(x\right)$ is shown above. What is $\underset{x\to 1}{lim}f\left(x\right)$?

Given the function $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ on the interval $(-1,2)$?

Which of the following best describes the remainder estimate for the integral test when determining the convergence of a series $\sum \underset{\infty }{n=1}{a}_n$ with $a_n=f\left(n\right)$, where $f$ is positive, continuous, and decreasing for $x\ge 1$?

If a function $f$ is continuous on $(-\infty ,\infty )$, which of the following statements is always true?

Which of the following functions is $o\left(x^2\right)$ as $x$ approaches $0$?

Which of the following series can be shown to converge by using the ratio test?

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3 f(x)

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $f\left(x\right)$ at a point $x=a$?