Functions, Graphs, and Limits
Now we'll check out one of the rock stars of the limit world:
To the right, to the right, to the right, to the right
We'll look at this limit one side at a time. First we'll look at the limit as x approaches 0 from the right:
We'll make a table to help us figure out what's going on.
Is there any number that gets close to as gets close to as x gets close to 0 from the right? Nope. keeps getting larger and larger and larger, without bound.
A function is bounded if there are a lower bound M and an upper bound N such that every value of f lies between M and N. If a function is unbounded, it may be missing an upper bound, missing a lower bound, or missing both. Talk about unlimited possibilities.
does not exist.
Most teachers are fine with saying
for this sort of limit, since the values of are getting larger and larger without bound as x approaches 0 from the right.
Be sure to find out before the test whether your teacher wants you to say ∞ or "does not exist'' for this sort of thing.
To the left, to the left, to the left, to the left
TGFT (thank goodness for tables)! They make things easier to understand.
As x approaches 0 from the left, is not approaching any particular number. Instead, is getting larger and more and more negative (technically is getting smaller while its magnitude is getting bigger). We could say
We can use table values to graph the function
As x gets closer to zero from the right, f(x) keeps growing like Pinocchio's nose. As x gets closer to zero from the left, f(x) keeps getting smaller and more negative. When we connect the dots with this in mind, we find this picture:
This line is an example of a vertical asymptote. The function gets ridiculously close to this line, since x keeps getting closer and closer to zero.
For those who would like a more definition-like definition, here it is. Let f(x) be some function, and a some value. If
is ∞ or -∞,
is ∞ or -∞,
we say f(x) has a vertical asymptote at x = a. On the graph this vertical asymptote is drawn as a dashed vertical line at x = a, and on at least one side of the vertical asymptote the function will be getting bigger and bigger (or more and more negative) as x approaches a.
Here are some of the things a function with a vertical asymptote at a could look like:
Now we'll talk about the pictures a bit. Picture 1 shows a function f(x) where
Picture 2 shows a function g(x) where
Picture 3 is the interesting one. Here,
and the function has a point where x = a. That is, h(a) is defined. It looks like the function is meeting the asymptote. While it might seem like this shouldn't be allowed, it's actually find.
Look at what happens as x approaches a from the right: the function gets more and more negative, but as long as x is greater than a, the function will not meet the asymptote.
The fact that f(a) exists and lies on the vertical asymptote line is fine.
On the other hand, we can't have something like this:
This is not a function, because f has infinitely many values when x = a, it fails the vertical line test.
A function can have one value on the vertical asymptote, but that's it.