Infinite Limits

The Division Problem

In arithmetic, you were taught that dividing by zero is illegal. It is "undefined." But in Calculus, we don't care about what happens at zero. We care about what happens when we get close to zero.

Let's look at the function $f(x) = 1/x$. Try plugging in smaller and smaller numbers:

As the denominator shrinks, the result grows. If we get infinitely close to zero, the result becomes infinitely large. We call this a Vertical Asymptote.

Formal Notation

Infinity ($\infty$) is not a number. You cannot hold it in your hand. It is a way of describing behavior. When we write:

We are saying: "As $x$ gets closer to 0, $f(x)$ gets larger than any number you can imagine." The line never hits a ceiling. It goes up forever.

One-Sided Behavior

Notice in the lab above (switch to $1/x$) that approaching from the left gives negative infinity ($-\infty$), while the right gives positive infinity $\infty$.

• Left Limit: $\lim_{x \to 0^-} \frac{1}{x} = -\infty$
• Right Limit: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$

Because the left and right sides don't agree (one goes up, one goes down), the general limit does not exist (DNE).