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Section 5.1 Infinity, limits, and power functions

In Section 3.2, we compared the behavior of the exponential functions \(p(t) = 2^t\) and \(q(t) = (\frac{1}{2})^t\text{,}\) and observed in Figure 3.2.5 that as \(t\) increases without bound, \(p(t)\) also increases without bound, while \(q(t)\) approaches \(0\) (while having its value be always positive). We also introduced shorthand notation for describing these phenomena, writing

\begin{equation*} p(t) \to \infty \text{ as } t \to \infty \end{equation*}

and

\begin{equation*} q(t) \to 0 \text{ as } t \to \infty\text{.} \end{equation*}

It's important to remember that infinity is not itself a number. We use the “\(\infty\)” symbol to represent a quantity that gets larger and larger with no bound on its growth.

We also know that the concept of infinity plays a key role in understanding the graphical behavior of functions. For instance, we've seen that for a function such as \(F(t) = 72 - 45e^{-0.05t}\text{,}\) \(F(t) \to 72\) as \(t \to \infty\text{,}\) since \(e^{-0.05t} \to 0\) as \(t\) increases without bound. The function \(F\) can be viewed as modeling the temperature of an object that is initially \(F(0) = 72-45 = 27\) degrees that eventually warms to \(72\) degrees. The line \(y = 72\) is thus a horizontal asymptote of the function \(F\text{.}\)

In Preview 5.1.1, we review some familiar functions and portions of their behavior that involve \(\infty\text{.}\)

Preview Activity 5.1.1.

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

  1. As \(t \to \infty\text{,}\) \(e^{-t} \to \) .

  2. As \(t \to \infty\text{,}\) \(\ln(t) \to \) .

  3. As \(t \to \infty\text{,}\) \(e^{t} \to \) .

  4. As \(t \to 0^+\text{,}\) \(e^{-t} \to \) . (When we write \(t \to 0^+\text{,}\) this means that we are letting \(t\) get closer and closer to \(0\text{,}\) but only allowing \(t\) to take on positive values.)

  5. As \(t \to \infty\text{,}\) \(35 + 53e^{-0.025t} \to\) .

  6. As \(t \to \frac{\pi}{2}^-\text{,}\) \(\tan(t) \to \). (When we write \(t \to \frac{\pi}{2}^-\text{,}\) this means that we are letting \(t\) get closer and closer to \(\frac{\pi}{2}^-\text{,}\) but only allowing \(t\) to take on values that lie to the left of \(\frac{\pi}{2}\text{.}\))

  7. As \(t \to \frac{\pi}{2}^+\text{,}\) \(\tan(t) \to \). (When we write \(t \to \frac{\pi}{2}^+\text{,}\) this means that we are letting \(t\) get closer and closer to \(\frac{\pi}{2}^+\text{,}\) but only allowing \(t\) to take on values that lie to the right of \(\frac{\pi}{2}\text{.}\))

Subsection 5.1.1 Limit notation

When observing a pattern in the values of a function that correspond to letting the inputs get closer and closer to a fixed value or letting the inputs increase or decrease without bound, we are often interested in the behavior of the function “in the limit”. In either case, we are considering an infinite collection of inputs that are themselves following a pattern, and we ask the question “how can we expect the function's output to behave if we continue?”

For instance, we have regularly observed that “as \(t \to \infty\text{,}\) \(e^{-t} \to 0\text{,}\)” which means that by allowing \(t\) to get bigger and bigger without bound, we can make \(e^{-t}\) get as close to \(0\) as we'd like (without \(e^{-t}\) ever equalling \(0\text{,}\) since \(e^{-t}\) is always positive).

Figure 5.1.1. Plots of \(y = e^t\) and \(y = e^{-t}\text{.}\)
Figure 5.1.2. Plots of \(y = e^t\) and \(y = \ln(t)\text{.}\)

Similarly, as seen in Figure 5.1.1 and Figure 5.1.2, we can make such observations as \(e^t \to \infty\) as \(t \to \infty\text{,}\) \(\ln(t) \to \infty\) as \(t \to \infty\text{,}\) and \(\ln(t) \to -\infty\) as \(t \to 0^+\text{.}\) We introduce formal limit notation in order to be able to express these patterns even more succinctly.

Definition 5.1.3.

Let \(L\) be a real number and \(f\) be a function. If we can make the value of \(f(t)\) as close to \(L\) as we want by letting \(t\) increase without bound, we write

\begin{equation*} \lim_{t \to \infty} f(t) = L \end{equation*}

and say that the limit of \(f\) as \(t\) increases without bound is \(L\).

If the value of \(f(t)\) increases without bound as \(t\) increases without bound, we instead write

\begin{equation*} \lim_{t \to \infty} f(t) = \infty\text{.} \end{equation*}

Finally, if \(f\) doesn't increase without bound, doesn't decrease without bound, and doesn't approach a single value \(L\) as \(t \to \infty\text{,}\) we say that \(f\) does not have a limit as \(t \to \infty\).

We use limit notation in related, natural ways to express patterns we see in function behavior. For instance, we write \(t \to -\infty\) when we let \(t\) decrease without bound, and \(f(t) \to -\infty\) if \(f\) decreases without bound. We can also think about an input value \(t\) approaching a value \(a\) at which the function \(f\) is not defined. As one example, we write

\begin{equation*} \lim_{t \to 0^+} \ln(t) = -\infty \end{equation*}

because the natural logarithm function decreases without bound as input values get closer and close to \(0\) (while always being positive), as seen in Figure 5.1.2.

In the situation where \(\lim_{t \to \infty} f(t) = L\text{,}\) this tells us that \(f\) has a horizontal asymptote at \(y = L\) since the function's value approaches this fixed number as \(t\) increases without bound. Similarly, if we can say that \(\lim_{t \to a} f(t) = \infty\text{,}\) this shows that \(f\) has a vertical asymptote at \(x = a\) since the function's value increases without bound as inputs approach the fixed number \(a\text{.}\)

For now, we are going to focus on the long-range behavior of certain basic, familiar functions and work to understand how they behave as the input increases or decreases without bound. Above we've used the input variable \(t\) in most of our previous work; going forward, we'll regularly use \(x\) as well.

Activity 5.1.2.

Complete the Table 5.1.4 by entering “\(\infty\text{,}\)” “\(-\infty\text{,}\)” “\(0\text{,}\)” or “no limit” to identify how the function behaves as either \(x\) increases or decreases without bound. As much as possible, work to decide the behavior without using a graphing utility.

Table 5.1.4. Some familiar functions and their limits as \(x \to \infty\) or \(x \to -\infty\text{.}\)
\(f(x)\) \(\lim_{x \to \infty} f(x)\) \(\lim_{x \to -\infty} f(x)\)
\(e^x\)    
\(e^{-x}\)    
\(\ln(x)\)    
\(x\)    
\(x^2\)    
\(x^3\)    
\(x^4\)    
\(\frac{1}{x}\)    
\(\frac{1}{x^2}\)    
\(\sin(x)\)    

Subsection 5.1.2 Power functions

To date, we have worked with several families of functions: linear functions of form \(y = mx + b\text{,}\) quadratic functions in standard form, \(y = ax^2 + bx + c\text{,}\) the sinusoidal (trigonometric) functions \(y = a\sin(k(x-b))+c\) or \(y = a\cos(k(x-b))+c\text{,}\) transformed exponential functions such as \(y = ae^{kx} + c\text{,}\) and transformed logarithmic functions of form \(y = a\ln(x) + c\text{.}\) For trigonometric, exponential, and logarithmic functions, it was essential that we first understood the behavior of the basic parent functions \(\sin(x)\text{,}\) \(\cos(x)\text{,}\) \(e^x\text{,}\) and \(\ln(x)\text{.}\) In order to build on our prior work with linear and quadratic functions, we now consider basic functions such as \(x\text{,}\) \(x^2\text{,}\) and additional powers of \(x\text{.}\)

Definition 5.1.5.

A function of the form \(f(x) = x^p\) where \(p\) is any real number is called a power function.

We first focus on the case where \(p\) is a natural number (that is, a positive whole number).

Activity 5.1.3.

Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zu. In what follows, we explore the behavior of power functions of the form \(y = x^n\) where \(n \ge 1\text{.}\)

  1. Press the “play” button next to the slider labeled “\(n\text{.}\)” 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.

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

  3. 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 \(-10,000 \le y \le 10,000\text{.}\) Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

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

In the situation where the power \(p\) is a negative integer (i.e., a negative whole number), power functions behave very differently. This is because of the property of exponents that states

\begin{equation*} x^{-n} = \frac{1}{x^n} \end{equation*}

so for a power function such as \(p(x) = x^{-2}\text{,}\) we can equivalently consider \(p(x) = \frac{1}{x^2}\text{.}\) Note well that for these functions, their domain is the set of all real numbers except \(x = 0\text{.}\) Like with power functions with positive whole number powers, we want to know how power functions with negative whole number powers behave as \(x\) increases without bound, as well as how the functions behave near \(x = 0\text{.}\)

Activity 5.1.4.

Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zv. In what follows, we explore the behavior of power functions \(y = x^n\) where \(n \le -1\text{.}\)

  1. Press the “play” button next to the slider labeled “\(n\text{.}\)” 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.

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

  3. How do your answers change on the interval \(0 \lt x \lt 1\text{?}\)

  4. 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 \(-10,000 \le y \le 10,000\text{.}\) Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

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

Subsection 5.1.3 Summary

  • The notation

    \begin{equation*} \lim_{x \to \infty} f(x) = L \end{equation*}

    means that we can make the value of \(f(x)\) as close to \(L\) as we'd like by letting \(x\) be sufficiently large. This indicates that the value of \(f\) eventually stops changing much and tends to a single value, and thus \(y = L\) is a horizontal asymptote of the function \(f\text{.}\)

    Similarly, the notation

    \begin{equation*} \lim_{x \to a} f(x) = \infty \end{equation*}

    means that we can make the value of \(f(x)\) as large as we'd like by letting \(x\) be sufficiently close, but not equal, to \(a\text{.}\) This unbounded behavior of \(f\) near a finite value \(a\) indicates that \(f\) has a vertical asymptote at \(x = a\text{.}\)

  • We summarize some key behavior of familiar basic functions with limits as \(x\) increases without bound in Table 5.1.6.

    Table 5.1.6. Some familiar functions and their limits as \(x \to \infty\) or \(x \to -\infty\text{.}\)
    \(f(x)\) \(\lim_{x \to \infty} f(x)\) \(\lim_{x \to -\infty} f(x)\)
    \(e^x\) \(\infty\) \(0\)
    \(e^{-x}\) \(0\) \(\infty\)
    \(\ln(x)\) \(\infty\) NA 1 
    \(x\) \(\infty\) \(-\infty\)
    \(x^2\) \(\infty\) \(\infty\)
    \(x^3\) \(\infty\) \(-\infty\)
    \(x^4\) \(\infty\) \(\infty\)
    \(\frac{1}{x}\) \(0\) \(0\)
    \(\frac{1}{x^2}\) \(0\) \(0\)
    \(\sin(x)\) no limit 2  no limit

    Additionally, Table 5.1.7 summarizes some key familiar function behavior where the function's output increases or decreases without bound as \(x\) approaches a fixed number not in the function's domain.

    Table 5.1.7. Some familiar functions and their limits involving \(\infty\) as \(x \to a\) where \(a\) is not in the function's domain.
    \(f(x)\) \(\lim_{x \to a^-} f(x)\) \(\lim_{x \to a^+} f(x)\)
    \(\ln(x)\) NA \(\lim_{x \to 0^+} \ln(x) = -\infty\)
    \(\frac{1}{x}\) \(\lim_{x \to 0^-} \frac{1}{x} = -\infty\) \(\lim_{x \to 0^+} \frac{1}{x} = \infty\)
    \(\frac{1}{x^2}\) \(\lim_{x \to 0^-} \frac{1}{x^2} = \infty\) \(\lim_{x \to 0^+} \frac{1}{x^2} = \infty\)
    \(\tan(x)\) \(\lim_{x \to \frac{\pi}{2}^-} \tan(x) = \infty\) \(\lim_{x \to \frac{\pi}{2}^+} \tan(x) = -\infty\)
    \(\sec(x)\) \(\lim_{x \to \frac{\pi}{2}^-} \sec(x) = \infty\) \(\lim_{x \to \frac{\pi}{2}^+} \sec(x) = -\infty\)
    \(\csc(x)\) \(\lim_{x \to 0^-} \sec(x) = -\infty\) \(\lim_{x \to 0^+} \sec(x) = \infty\)
  • A power function is a function of the form \(f(x) = x^p\) where \(p\) is any real number. For the two cases where \(p\) is a positive whole number or a negative whole number, it is straightforward to summarize key trends in power functions' behavior.

    • If \(p = 1, 2, 3, \ldots\text{,}\) then the domain of \(f(x) = x^p\) is the set of all real numbers, and as \(x \to \infty\text{,}\) \(f(x) \to \infty\text{.}\) For the limit as \(x \to -\infty\text{,}\) it matters whether \(p\) is even or odd: if \(p\) is even, \(f(x) \to \infty\) as \(x \to -\infty\text{;}\) if \(p\) is odd, \(f(x) \to -\infty\) as \(x \to \infty\text{.}\) Informally, all power functions of form \(f(x) = x^p\) where \(p\) is a positive even number are “U-shaped”, while all power functions of form \(f(x) = x^p\) where \(p\) is a positive odd number are “chair-shaped”.

    • If \(p = -1, -2, -3, \ldots\text{,}\) then the domain of \(f(x) = x^p\) is the set of all real numbers except \(x=0\text{,}\) and as \(x \to \pm \infty\text{,}\) \(f(x) \to 0\text{.}\) This means that each such power function with a negative whole number exponent has a horizontal asymptote of \(y = 0\text{.}\) Regardless of the value of \(p\) (\(p = -1, -2, -3, \ldots\)), \(\lim_{x \to 0^+} f(x) = \infty\text{.}\) But when we approach \(0\) from the negative side, it matters whether \(p\) is even or odd: if \(p\) is even, \(f(x) \to \infty\) as \(x \to 0^-\text{;}\) if \(p\) is odd, \(f(x) \to -\infty\) as \(x \to 0^-\text{.}\) Informally, all power functions of form \(f(x) = x^p\) where \(p\) is a negative odd number look similar to \(\frac{1}{x}\text{,}\) while all power functions of form \(f(x) = x^p\) where \(p\) is a negative even number look similar to \(\frac{1}{x^2}\text{.}\)

Because the domain of the natural logarithm function is only positive real numbers, it doesn't make sense to even consider this limit.
Because the sine function neither increases without bound nor approaches a single value, but rather keeps oscillating through every value between \(-1\) and \(1\) repeatedly, the sine function does not have a limit as \(x \to \infty\text{.}\)

Exercises 5.1.4 Exercises

1.
2.
3.
4.

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

  1. Use a computational device to compute decimal expressions for \(f(10)\text{,}\) \(g(10)\text{,}\) and \(h(10)\text{,}\) as well as \(f(100)\text{,}\) \(g(100)\text{,}\) and \(h(100)\text{.}\) What do you observe?

  2. For each of \(f\text{,}\) \(g\text{,}\) and \(h\text{,}\) how large an input is needed in order to ensure that the function's output value is at least \(10^{10}\text{?}\) What do these values tell us about how each function grows?

  3. Consider the new function \(r(x) = \frac{g(x)}{h(x)} = \frac{x^2}{e^x}\text{.}\) Compute \(r(10)\text{,}\) \(r(100)\text{,}\) and \(r(1000)\text{.}\) What do the results suggest about the long-range behavior of \(r\text{?}\) What is surprising about this, in light of the fact that both \(x^2\) and \(e^x\) grow without bound?

5.

Consider the familiar graph of \(f(x) = \frac{1}{x}\text{,}\) which has a vertical asypmtote at \(x = 0\) and a horizontal asymptote at \(y = 0\text{,}\) as pictured in Figure 5.1.8. In addition, consider the similarly-shaped function \(g\) shown in Figure 5.1.9, which has vertical asymptote \(x = -1\) and horizontal asymptote \(y = -2\text{.}\)

Figure 5.1.8. A plot of \(y = f(x) = \frac{1}{x}\text{.}\)
Figure 5.1.9. A plot of a related function \(y = g(x)\text{.}\)
  1. How can we view \(g\) as a transformation of \(f\text{?}\) Explain, and state how \(g\) can be expressed algebraically in terms of \(f\text{.}\)

  2. Find a formula for \(g\) as a function of \(x\text{.}\) What is the domain of \(g\text{?}\)

  3. Explain algebraically (using the form of \(g\) from (b)) why \(\lim_{x \to \infty} g(x) = -2\) and \(\lim_{x \to -1^+} g(x) = \infty\text{.}\)

  4. What if a function \(h\) (again of a similar shape as \(f\)) has vertical asymptote \(x = 5\) and horizontal asymtote \(y = 10\text{.}\) What is a possible formula for \(h(x)\text{?}\)

  5. Suppose that \(r(x) = \frac{1}{x+35} - 27\text{.}\) 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?

6.

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

  1. Compare and contrast the graphs of \(f\text{,}\) \(g\text{,}\) and \(h\text{.}\) How are they similar? How are they different? (There is a lot you can discuss here.)

  2. Observe that we can think of \(f(x) = x^{2.4}\) as \(f(x) = x^{24/10} = x^{12/5}\text{.}\) In addition, recall by exponent rules that we can also view \(f\) as having the form \(f(x) = \sqrt[5]{x^{12}}\text{.}\) Write \(g\) and \(h\) in similar forms, and explain why \(g\) has a different domain than \(f\) and \(h\text{.}\)

  3. How do the graphs of \(f\text{,}\) \(g\text{,}\) and \(h\) compare to the graphs of \(y = x^2\) and \(y = x^3\text{?}\) Why are these natural functions to use for comparison?

  4. Explore similar questions for the graphs of \(p(x) = x^{-2.4}\text{,}\) \(q(x) = x^{-2.5}\text{,}\) and \(r(x) = x^{-2.6}\text{.}\)