- Topics At a Glance
- Limits
- Functions Are Your Friend
- Graphing and Visualizing Limits
- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
**All About Asymptotes****Vertical Asymptotes**- Finding Vertical Asymptotes
- Vertical Asymptotes vs. Holes
- Limits at Infinity
- Natural Numbers
- Limits Approaching Zero
- Estimating a Circle
- The Cantor Set and Fractals
- Limits of Functions at Infinity
- Horizontal, Slant, and Curvilinear Asymptotes
- Horizontal Asymptoes
- Slant Asymptotes
- Curvilinear Asymptotes
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Comparing Functions
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
- Manipulating Limits
- The Basic Properties
- Multiplication by a Constant
- Adding and Subtracting Limits
- Multiplying and Dividing Limits
- Powers and Roots of Limits
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Now we'll check out one of the rock stars of the limit world:

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.

The limit

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.

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

or

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 -∞,

or if

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.

Exercise 1

PICTURE: "Which graphs show functions with asymptotes at..."