5.1 Infinity, limits, and power functions

Introduction

Limit notation

Power functions

Summary

Complete each of the following statements with an appropriate number or the symbols $\infty$ or $-\infty$. Do your best to do so without using a graphing utility; instead use your understanding of the function's graph.

As $t \to \infty$, $e^{-t} \to$.

Press the “play” button next to the slider labeled “$n$.” Watch at least two loops of the animation and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

Click the icons next to each of the following 8 functions so that you can see all of $y = x$, $y = x^2$, $\ldots$, $y = x^8$ graphed at once. On the interval $0 \lt x \lt 1$, how do the graphs of $x^a$ and $x^b$ compare if $a \lt b$?

Uncheck the icons on each of the 8 functions to hide their graphs. Click the settings icon to change the domain settings for the axes, and change them to $-10 \le x \le 10$ and $-10,000 \le y \le 10,000$. Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

Click the icons next to each of the following 8 functions so that you can see all of $y = x$, $y = x^2$, $\ldots$, $y = x^8$ graphed at once. On the interval $x \gt 1$, how do the graphs of $x^a$ and $x^b$ compare if $a \lt b$?

Press the “play” button next to the slider labeled “$n$.” Watch two loops of the animation and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

Click the icons next to each of the following 8 functions so that you can see all of $y = x^{-1}$, $y = x^{-2}$, $\ldots$, $y = x^{-8}$ graphed at once. On the interval $1 \lt x$, how do the functions $x^a$ and $x^b$ compare if $a \lt b$? (Be careful with negative numbers here: e.g., $-3 \lt -2$.)

How do your answers change on the interval $0 \lt x \lt 1$?

Uncheck the icons on each of the 8 functions to hide their graphs. Click the settings icon to change the domain settings for the axes, and change them to $-10 \le x \le 10$ and $-10,000 \le y \le 10,000$. Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

Explain why $\lim_{x \to \infty} \frac{1}{x^n} = 0$ for any choice of $n = 1, 2, \ldots$.

We've observed that several different familiar functions grow without bound as $x \to \infty$, including $f(x) = \ln(x)$, $g(x) = x^2$, and $h(x) = e^x$. In this exercise, we compare and contrast how these three functions grow.

• Use a computational device to compute decimal expressions for $f(10)$, $g(10)$, and $h(10)$, as well as $f(100)$, $g(100)$, and $h(100)$. What do you observe? - For each of $f$, $g$, and $h$, how large an input is needed in order to ensure that the function's output value is at least $10^{10}$? What do these values tell us about how each function grows? - Consider the new function $r(x) = \frac{g(x)}{h(x)} = \frac{x^2}{e^x}$. Compute $r(10)$, $r(100)$, and $r(1000)$. What do the results suggest about the long-range behavior of $r$? What is surprising about this, in light of the fact that both $x^2$ and $e^x$ grow without bound?

Consider the familiar graph of $f(x) = \frac{1}{x}$, which has a vertical asypmtote at $x = 0$ and a horizontal asymptote at $y = 0$, as pictured in Figure. In addition, consider the similarly-shaped function $g$ shown in Figure, which has vertical asymptote $x = -1$ and horizontal asymptote $y = -2$.

• How can we view $g$ as a transformation of $f$? Explain, and state how $g$ can be expressed algebraically in terms of $f$. - Find a formula for $g$ as a function of $x$. What is the domain of $g$? - Explain algebraically (using the form of $g$ from (b)) why $\lim_{x \to \infty} g(x) = -2$ and $\lim_{x \to -1^+} g(x) = \infty$. - What if a function $h$ (again of a similar shape as $f$) has vertical asymptote $x = 5$ and horizontal asymtote $y = 10$. What is a possible formula for $h(x)$? - Suppose that $r(x) = \frac{1}{x+35} - 27$. Without using a graphing utility, how do you expect the graph of $r$ to appear? Does it have a horizontal asymptote? A vertical asymptote? What is its domain?

Power functions can have powers that are not whole numbers. For instance, we can consider such functions as $f(x)=x^{2.4}$, $g(x)=x^{2.5}$, and $h(x)=x^{2.6}$.

• Compare and contrast the graphs of $f$, $g$, and $h$. How are they similar? How are they different? (There is a lot you can discuss here.) - Observe that we can think of $f(x) = x^{2.4}$ as $f(x) = x^{24/10} = x^{12/5}$. In addition, recall by exponent rules that we can also view $f$ as having the form $f(x) = \sqrt[5]{x^{12}}$. Write $g$ and $h$ in similar forms, and explain why $g$ has a different domain than $f$ and $h$. - How do the graphs of $f$, $g$, and $h$ compare to the graphs of $y = x^2$ and $y = x^3$? Why are these natural functions to use for comparison? - Explore similar questions for the graphs of $p(x) = x^{-2.4}$, $q(x) = x^{-2.5}$, and $r(x) = x^{-2.6}$.