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Calculus Done Right

# Infinite Limits

In this quiz, we will explore two types of “infinite limits.” The first type is a bit of a misnomer: in fact, it is an example of a way in which a limit can fail to exist.

Let $$f(x)$$ be a function and $$a$$ be some real number. We say that $$\lim\limits_{x\to a} f(x) = \infty$$ if $$f(x)$$ “blows up” as $$x$$ gets near $$a$$; that is, for all $$x$$ sufficiently close to $$a$$ (but not equal to $$a$$), $$f(x)$$ gets arbitrarily large. When this happens, even though we will write $$\lim\limits_{x\to a} f(x) = \infty,$$ we'll still say the limit doesn't exist!

Here’s one way to see what this means: suppose we pick any large number, say $$100000.$$ Then there is some interval around $$a$$ such that $$f(x) > 100000$$ for all the $$x$$ in this interval (except possibly for $$x=a$$).

One way to investigate this is by plugging in numbers closer and closer to $$a,$$ and seeing that they get larger and larger without bound.

Similarly, $$\lim\limits_{x\to a} f(x) = -\infty$$ if $$f(x)$$ “blows up” in a downward direction: for all $$x$$ sufficiently close to $$a$$ (but not equal to $$a$$), $$f(x)$$ gets arbitrarily large and negative (i.e. $$-f(x)$$ gets arbitrarily large).

Let’s see what $$\lim\limits_{x\to a} f(x) = \infty$$ looks like. Which of these is the graph of a function $$f(x)$$ such that $$\lim\limits_{x\to 0} f(x) = \infty$$?

When a function is given by a formula instead of a graph, how do we determine if $$\lim\limits_{x\to a} f(x) = \infty$$? We imagine plugging in numbers close to $$a$$, and analyze what happens to the expression. We don’t have to actually plug numbers into a calculator; we can just analyze what would happen if $$x$$ were close to $$a.$$

For instance, look at $$\dfrac{x+2}{(x-1)^2}$$ near $$x=1.$$ When $$x$$ is close to 1, the denominator is a very small positive number, while the numerator is close to 3. So the whole thing blows up, and so $$\lim\limits_{x\to 1} \dfrac{x+2}{(x-1)^2} = \infty$$. Be careful though: If both the numerator and denominator are very small or very large, we have an indeterminate form, and we can’t conclude anything without further work!

Which of these functions satisfies $$\lim\limits_{x\to 1} f(x) = \infty$$?

Remember that writing $$\lim\limits_{x\to a} f(x) = \infty$$ is a shorthand for “the limit does not exist because the function blows up at $$a.$$” This shorthand can be useful, but it’s very important not to manipulate it as if $$\infty$$ is a number. Consider the following three statements:

• “$$\infty + \infty = \infty$$”: if $$\lim\limits_{x\to a} f(x) = \infty$$ and $$\lim\limits_{x\to a} g(x) = \infty,$$ then $$\lim\limits_{x\to a} \left[f(x) + g(x)\right] = \infty.$$
• “$$\infty - \infty = 0$$”: if $$\lim\limits_{x\to a} f(x) = \infty$$ and $$\lim\limits_{x\to a} g(x) = \infty,$$ then $$\lim\limits_{x\to a} \left[f(x) - g(x)\right] = 0.$$
• “$$\infty \cdot \infty = \infty$$”: if $$\lim\limits_{x\to a} f(x) = \infty$$ and $$\lim\limits_{x\to a} g(x) = \infty,$$ then $$\lim\limits_{x\to a} \left[f(x) \cdot g(x)\right] = \infty.$$

How many of these are true?

The second type of infinite limit is the limit “at infinity”: instead of looking at places where a function $$f(x)$$ gets arbitrarily large, we look at what happens to $$f(x)$$ as $$x$$ gets arbitrarily large. That is, we say that $$\lim\limits_{x\to\infty} f(x) = L$$ if $$f(x)$$ gets arbitrarily close to $$L$$ as $$x$$ “approaches $$\infty,$$” i.e. gets arbitrarily large.

So if you pick your favorite small positive number, say $$0.001,$$ then $$f(x)$$ is within $$0.001$$ of $$L$$ for all sufficiently large $$x$$ (i.e. for all $$x$$ in an “interval around $$\infty$$”).

If you picked a large enough $$a,$$ you would find that f(x) is within 0.001 of 0 for all $$x>a.$$

As with infinite limits, one way to investigate this is by plugging larger and larger values of $$x$$ and seeing that $$f(x)$$ gets closer and closer to $$L.$$

Similarly we can define $$\lim\limits_{x\to -\infty} f(x)$$ by looking at what happens to $$f(x)$$ as $$x$$ gets arbitrarily large in the negative direction, i.e. farther and farther to the left on the number line.

Now that we know what the limit of $$f(x)$$ at infinity means, let’s see what it says about the graph of $$f(x).$$

Which of the following is the graph of a function $$f(x)$$ that satisfies $$\lim\limits_{x\to\infty} f(x) = 1$$?

Which of these satisfies $$\lim\limits_{x\to-\infty} f(x) = 0$$?

One family of limits at infinity you will get to know are the reciprocal power functions, like $$\frac{1}{x}$$ and $$\frac{1}{x^2}.$$ Clearly, $$\lim\limits_{x \to \infty} \frac{1}{x} = 0,$$ because as $$x$$ gets very large, its reciprocal gets very small. Similarly for $$\frac{1}{x^2}.$$

What about in general? For what values of $$p$$ is it the case that

$\lim_{x \to \infty} \frac{1}{x^p} = 0\ ?$

You will have to remind yourself what negative and fractional exponents mean!

Consider the following statement:

If $$\lim\limits_{x\to\infty} f(x) = 0,$$ then for large enough $$x,$$ $$f(x)$$ gets closer and closer to $$0$$ without ever equaling $$0.$$

Which of the following choices is correct?

A) The statement is true.

B) The statement is false, and $$f(x) = \dfrac{1}{x}$$ is a counterexample.

C) The statement is false, and $$f(x) = \dfrac{\ln(x)}{x}$$ is a counterexample.

D) The statement is false, and $$f(x) = \dfrac{\sin(x)}{x}$$ is a counterexample.

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