Difference between infinite and undefined

Question

In questions, sometimes I see these terms being used interchangeably. But also some lecturers say that they are different. I am pretty confused with these. Please explain.

Answer 1

Consider the function $f(x)=\frac{1}{x}$.

This function is not defined in $x=0$, because what number could $\frac{1}{0}$ be equal to? If we take progressively smaller values for $x$, e.g. $0.1, 0.01, 0.001,...$ it is obvious that $f(x)$ gets larger and larger. But, what is the largest number? There isn't one, because for any given large real number, we can find an even larger real number.

Hence, we are prompted to consider the limit: $$\lim\limits_{x\to0} \frac{1}{x}$$

Does this limit exist?

No, because we can say the following. However small $x$ becomes, $\frac{1}{x}$ gets bigger and bigger in absolute value. But the problem is that depending on whether $x<0$ or $x>0$, $\frac{1}{x}$ can go either to $-\infty$ or $+\infty$. Since the limit of a function (if it exists) must be unique, we conclude that this limit cannot exist.

How could we fix this?

We can get rid of the dependency on the sign of $f(x)$. Instead of looking at the limit as defined above, we can look at the one-sided limits $$\lim\limits_{x\to0^-}f(x) \text{ and } \lim\limits_{x\to0^+} f(x)$$ Now, both of these limits exist and they are $-\infty$ and $+\infty$ respectively. We can still say that these limits do not exist, in the sense that they do not exist in $\mathbb R$, since $\infty$ is not a real number. Us saying that a limit is infinite is just a way to simplify the notation of the fact that a function does not converge to a specific (finite) value.

Another example of when a limit does not exist would be the function $g(x)=\sin x$ and the limit $$\lim\limits_{x\to+\infty} g(x)$$ This limit does not exist because however large $x$ becomes, the values of $g(x)$ do not get closer and closer to a specific value. Instead, they always oscillate back and forth, taking various values between -1 and 1.

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There are of course formal proofs of the claims I have made, and they are not very difficult to understand, but I think that this is gives a good intuition on what it means when something is defined or not.

Answer 2

If you're in the context of limits, then you should think of "infinite" as a special kind of "undefined."

A limit can be undefined for a lot of reasons. For instance

$$\lim_{x \to \infty} \sin x$$

is undefined because the limit doesn't get close and stay close to any particular number.

But the limit

$$\lim_{x \to 0} \frac{1}{x^2}$$

is undefined for a special reason: The value of the function increases without bound. So we write,

$$\lim_{x \to 0} \frac{1}{x^2}= +\infty$$

to say that this limit is undefined, because the function goes to positive infinity.

Note that, by contrast, the limit

$$\lim_{x \to 0} \frac{1}{x}$$

is not infinity, because the limit from the left increases without bound, but the limit on the right DEcreases without bound. So it's not the same as the previous example. It's "just plain" undefined, not "undefined because it goes to infinity."