1. Limits and Continuity

Which of the following is true about the graph of $y=\ln |x^2-1|$ in the interval $(-1,1)$?

Estimate the limit $\underset{n\to \infty }{lim}{(2n+1)}^{-9}$ correct to five decimal places.

Given that $f^{\left(n\right)}\left(0\right)$=$(n+1)!$ for $n=0,1,2,\dots$, which of the following is the Maclaurin series for $f\left(x\right)$?

Suppose the average rate of change of $g\left(x\right)$ between $x=4$ and $x=7$ is $0$. Which statement must be true?

Given a table of values for a function $f(x,y)$, which of the following best describes how to estimate the mixed partial derivative $f_{xy}(3,2)$?

Given $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ over the interval $[4,6]$?

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

Determine the interval of convergence for the series $\sum n_1\infty \frac{{(x-2)}^n}{n}$. Express your answer using interval notation.

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 $\underset{x\to 1}{lim}f\left(x\right)$?