Make another table for \(1/x\) but this time with value of \(x\) approaching \(0\) from the left. What do you see? What is a reasonable way to use limit notation and the \(\infty\) symbol to express what you find?
Section 1.2 Infinity Stuff
Consider the function \(f(x)=1/x\text{.}\) What can you say about \(\displaystyle \lim_{x\to 0^{+}}f(x)\text{?}\) Well, let’s make a table and see.
| \(x\) | \(1/x\) |
|---|---|
| 1.000 | 1 |
| 0.100 | 10 |
| 0.010 | 100 |
| 0.001 | 1000 |
So, it’s pretty apparent that as \(x\to 0^{+}\text{,}\) the function \(1/x\) isn’t headed toward any number. So, there is no limit. Said differently, the right hand limit of \(1/x\)does not exist at \(0\text{.}\) So, the limit notation we’ve created so far isn’t useful. However, it would be useful to record this information. To do that, we write \(\displaystyle \lim_{x\to 0^{+}}f(x)=\infty\text{.}\) Please note here that this notation is *not* saying that a limit exists. It just is language explaining *how* it doesn’t exist.
Example 1.2.1.
Definition 1.2.2.
When \(f(x)\) grows positively without bound as \(x\) goes to \(a\) from the right, we write \(\displaystyle \lim_{x\to a^{+}}f(x)=\infty\text{.}\) Also, when \(f(x)\) grows positively without bound as \(x\) goes to \(a\) from the left, we write \(\displaystyle \lim_{x\to a^{-}}f(x)=\infty\text{.}\) Similarly, when \(f(x)\) grows negatively without bound as \(x\) goes to \(a\) from the right, we write \(\displaystyle \lim_{x\to a^{+}}f(x)=-\infty\) and when \(f(x)\) grows negatively without bound as \(x\) goes to \(a\) from the left, we write \(\displaystyle \lim_{x\to a^{-}}f(x)=-\infty\text{.}\) As with actual limits, if we omit the \(+\) and \(-\) then we are saying that the behavior is the same from both directions.
We also use the \(\infty\) symbol as a *place* to let the domain variable *approach*. When we let \(x\) get arbitrarily positively large, we write \(x \to \infty\) So, for example, consider our friend \(1/x\text{.}\) If you make a table for this taking larger and larger positive values of \(x\text{,}\) you’ll see that \(1/x\) approaches \(0\text{.}\) So, in that case, we can write \(\displaystyle \lim_{x\to \infty}1/x=0\text{.}\)
Example 1.2.3.
Make another table for \(1/x\) but this time with value of \(x\) getting negatively large without bound (e.g. \(-1, -10, -100, -1000, etc.\)). What do you see? What is a reasonable way to use limit notation and the \(\infty\) symbol to express what you find?
Definition 1.2.4.
To express a limit when we let the domain variable grow positively without bound, we write \(\displaystyle \lim_{x\to \infty}\) and to express a limit when we let the domain variable grow negatively without bound, we write \(\displaystyle \lim_{x\to -\infty}\text{.}\)
Example 1.2.5.
Use limit notation and the infinity symbol to express what is happening to \(x^2\) as \(x\) grows positively without bound.
