Chapter 2

Calculate the following limits using the factorization formula $x^n-a^n=(x-a)(x^{n-1}+a{x}^{n-2}+a^2{x}^{n-3}+\cdots +a^{n-2}x+a^{n-1})$ where $n$ is a positive integer and a is a real number. $$\underset{x\to -1}{lim}\frac{x^7+1}{x+1}$$ (Hint: Use the formula for $x^7-a^7$ with $a=-1$.)

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let $f(t)$ be your distance from the car $t$ hours after 7 A.M. on Friday morning, and let $g(t)$ be your distance from the car $t$ hours after 7 A.M. on Sunday morning. a. Evaluate $f(0),f(2),g(0),$ and $g(2)$ b. Let $h(t)=f(t)-g(t).$ Find $h(0)$ and $h(2)$ c. Show that there is some point along the trail that you will pass at exactly the same time on both days.

End behavior of rational functions Suppose $f(x)=\frac{p(x)}{q(x)}$ is a rational function, where $p(x)=a_m{x}^m+a_{m-1}{x}^{m-1}+\cdots +a_2{x}^2+a_{1}x+a_0$ $q(x)=b_n{x}^n+b_{n-1}{x}^{n-1}+\cdots +b_2{x}^2+b_{1}x+b_0,a_m\ne 0$ and $b_n\ne 0$. a. Prove that if $m=n,$ then $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)=\frac{a_m}{b_n}$ b. Prove that if \(m

Calculate the following limits using the factorization formula $x^n-a^n=(x-a)(x^{n-1}+a{x}^{n-2}+a^2{x}^{n-3}+\cdots +a^{n-2}x+a^{n-1})$ where $n$ is a positive integer and a is a real number. $$\underset{x\to a}{lim}\frac{x^5-a^5}{x-a}$$

Horizontal and slant asymptotes a. Is it possible for a rational function to have both slant and horizontal asymptotes? Explain. b. Is it possible for an algebraic function to have two distinct slant asymptotes? Explain or give an example.

A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and the descent? Explain. (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation)

Does continuity of $|\mathit{f}|$ imply continuity of $\mathit{f}$ ? Let $$g(x)=\left\{$$\begin{array}{ll}1& \text{ if }x\ge 0\\ \\ -1& \text{ if }x<0\end{array}$$\right.$$ a. Write a formula for $|g(x)|$ b. Is $g$ continuous at $x=0?$ Explain. c. Is $|g|$ continuous at $x=0?$ Explain. d. For any function $f$, if $|f|$ is continuous at $a$, does it necessarily follow that $f$ is continuous at $a$ ? Explain.

End behavior of exponentials Use the following instructions to determine the end behavior of $f(x)=\frac{4e^x+2e^{2x}}{8e^x+e^{2x}}$ a. Evaluate lim $f(x)$ by first dividing the numerator and denominator by $e^{2x}$ b. Evaluate $\underset{x\to -\mathrm{\infty }}{lim}f(x)$ by first dividing the numerator and denominator by $e^x$ c. Give the horizontal asymptote(s). d. Graph $f$ to confirm your work in parts (a)-(c).

Slope of a tangent line a. Sketch a graph of $y=2^x$ and carefully draw three secant lines connecting the points $P(0,1)$ and $Q(x,2^x),$ for $x=-3,-2,$ and -1 b. Find the slope of the line that passes through $P(0,1)$ and $Q(x,2^x),$ for $x\ne 0$. c. Complete the table and make a conjecture about the value of $\underset{x\to 0^{-}}{lim}\frac{2^x-1}{x}$. $$\begin{array}{|lllllll|}\hline x& -1& -0.1& -0.01& -0.001& -0.0001& -0.00001\\ \frac{2^x-1}{x}& & & & & & \\ \hline\end{array}$$

Find the horizontal asymptotes of each function using limits at infinity. $$f(x)=\frac{2e^x+3}{e^x+1}$$