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C.4 Some Limits at Negative Infinity

Let’s now examine the limits at negative infinity of some common functions. We’ll again use epsilon-strips to assist our thinking, and we’ll move faster than we did on the preceding screen since the reasoning is quite similar.


The limit at negative infinity exists and equals L if,
for any value of $\epsilon \gt 0$ that we choose,
there is a value of M such that for all $x \lt M$
the function’s output values lie in the range $L\, – \epsilon \lt f(x) \lt L + \epsilon.$

Note: When looking at functions like $x^n,$ we’re going to restrict ourselves to integer values of n so we don’t have to worry about the nth-root (like $\sqrt{x} = x^{0.5}$) of negative numbers, since we’re now looking at $x \to -\infty.$

$\color{blue}{\text{I. }f(x) = \sin(x)}$

What is $\displaystyle{\lim_{x \to -\infty} \sin(x) }?$ How would you explain your conclusion to a friend?

Graph of $f(x) = \sin(x)$ versus x


The reasoning and conclusion for what happens as $x \to -\infty$ is the same as that for $x \to \infty$ from the preceding screen:

Conclusion: $\displaystyle{\lim_{x \to -\infty} \sin(x) = \text{ DNE} \, \cmark},$ because there is no value of L for which this works given our chosen $\epsilon = 0.5$ or $=0.1.$
Similarly, $\displaystyle{\lim_{x \to -\infty} \cos(x) = \text{ DNE} \, \cmark}.$

$\color{blue}{\text{II. }f(x) = e^x}$

Does $\displaystyle{\lim_{x \to -\infty} e^x }$ exist? If so, what is its value?

Graph of $f(x) = e^x$ versus x


As you can see, only the value $L = 0$ satisfies the condition $L – \epsilon \lt f(x) \lt L + \epsilon$ for any $\epsilon \gt 0.$

Conclusion: $\displaystyle{\lim_{x \to -\infty}e^x = 0 \, \cmark}.$

$\color{blue}{\text{III. }f(x) = x^n,\; \text{integer } n \ge 1}$

What is $\displaystyle{\lim_{x \to -\infty} x^n }$ for integer values of $n \ge 1$?
You can use the slider beneath the graph to change n. As you try different values, what do you notice about what happens as $x \to -\infty$ for odd versus even values of n?

Graph of $f(x) = x^n$ versus x


Currently n = 3: $f(x) = x^3$

Conclusion: $\displaystyle{\lim_{x \to -\infty} x^n } = \text{ DNE} \, \cmark$ for integer $n \ge 1.$ If we choose to convey more information about the way in which the limit does not exist as $x \to -\infty$, we can further specify that
\[ \lim_{x \to -\infty} x^n =
\infty, & \text{if $n$ is even} \\[8px] -\infty, & \text{if $n$ is odd}

$\color{blue}{\text{IV. }f(x) = C, \text{constant function}}$

Consider the function $f(x) = C,$ where C is a constant (say, 42).
$\displaystyle{\lim_{x \to -\infty} C = ? }$

Graph of $f(x)=42$ versus x


Conclusion: Let C be a constant value. Then $\displaystyle{\lim_{x \to -\infty}C = C \, \cmark}$

$\color{blue}{\text{V. }f(x) = x^{-n} = \dfrac{1}{x^n}, \; \text{integer } n \ge 1}$

For $n \ge 1,$ $\displaystyle{\lim_{x \to -\infty}x^{-n} = \lim_{x \to -\infty} \dfrac{1}{x^n} = ? }$

Graph of $f(x) = x^{-n} = \dfrac{1}{x^n}$ versus x


Currently n = 2: $f(x) = x^{-2} = \dfrac{1}{x^2}$

Conclusion: For $n \ge 1,$ $\displaystyle{\lim_{x \to -\infty}x^{-n} = \lim_{x \to -\infty} \dfrac{1}{x^n} = 0 \, \cmark }$

Are you finding the epsilon strip a helpful tool? Do you have other functions in mind where the epsilon strip helps you decide whether the limit at infinity exists or not? Join the discussion over on the Forum and let us know!

We know that the above is pretty quick reasoning about whether the various limits exist, but that’s the point: with the epsilon strip in mind, we can quickly reach the conclusion we need to. To be clear, we are not proving that these limits do or do not exist, but for our Introductory Calculus purposes, this will suffice. (And if you want to explore how to actually prove our conclusions above, we again encourage you to investigate an upper division Mathematics course in “Analysis,” which is devoted to such things!)

On the next screen we’ll use the conclusions above to lay some groundwork for deciding determining the limit as $x \to \pm\infty$ for many, many more functions.

The Upshot

We’re going to essentially repeat the Upshot from the preceding screen since it still applies:

  1. While it’s handy to keep the conclusions above in mind as we proceed, please don’t work to memorize these results. Instead practice visualizing the epsilon strip, and mentally overlay that on top of a graph of any function you’re given to determine whether the limit at (negative) infinity exists.