1. Limits and Continuity

Given that the definite integral from $0$ to $\pi$ of ${\sin }^4\left(x\right)$ $dx$ equals $\frac{3\pi }{8}$, what is the value of the definite integral from $0$ to $\pi$ of ${\sin }^4(\pi -x)$ $dx$?

Given the double integral $\int {\int }_{y=0}{y=2}^{}{\int }_{x=y^2}{x=4}^{}f(x,y) dx dy$, which of the following represents the correct limits of integration after changing the order of integration?

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

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. 

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

Given the graph of the function $g\left(x\right)$ below, on which interval is $g\left(x\right)$ continuous?

Which of the following regions in the plane has an area equal to the limit $\underset{n\to \infty }{lim}\frac{1}{n}(1+\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\dots +\frac{1}{n^{n-1}})$? Do not evaluate the limit.

Given the geometric series $\sum n_0=\infty x^n$, find the sum of the series $\sum n_1=\infty n{x}^{n-1}$, where $\left|x\right|<1$.

Use the ratio test to determine whether the series $\sum _{n=1}^{\infty }\frac{\left(n\right)!}{n^n}$ converges or diverges.

Given the graph of a function $f$, find a number $\epsilon >0$ such that if $|x-1|<\epsilon$, then $\left|f\right(x)-1|<0.2$.

For the function $f\left(x\right)=2x^2+3x$, what is the average rate of change from $x=0$ to $x=5$?

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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Which of the following statements about the series $\sum \underset{}{n=1}\infty \frac{1}{2^n}$ is true?

Consider the function $f\left(x\right)=\{\begin{array}{cc}{x}^2& \text{if}x<2\\ 5& \text{if}x=2\\ x+2& \text{if}x>2\end{array}$. Determine if $f\left(x\right)$ has any points of discontinuity, and explain your reasoning.

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 4 + 3x² / (x² + 1)

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x