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Some things are untouchable. The Mona Lisa, MC Hammer...asymptotes?
An asymptote is a line to which a function gets infinitely close. There are several different types of asymptotes, and it will be our pleasure to introduce each and every one of them.
If an asymptote is present, it will exist in at least one infinite direction, but it's actually possible for the actual asymptote line to be intersected at some point.
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 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. In less mathy terms, the function can't get as big as it wants to. 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.
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 x = 0 is an example of a vertical asymptote. The function gets ridiculously close to this line, as 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 fine.
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 totes okay.
Vertical asymptotes most frequently show up in rational functions. When a rational function, f(x), has a non-zero constant in the numerator and an expression with a variable in the denominator, the function f(x) will have vertical asymptotes at all values of x that make the denominator 0. If the denominator has no roots, then f(x) will have no vertical asymptotes.This works for the same reason that 1⁄x has a vertical asymptote at zero: the numerator is a non-zero constant and the denominator is getting smaller and smaller, therefore the fraction will get bigger and bigger.
If we use a different constant, the principle is still the same.
If a rational function has something besides a non-zero constant in the numerator, we may need to be creative: factor the numerator and denominator, cancel and common factors, and find the roots of whatever polynomial remains in the denominator after factoring.
Be Careful: Remember to simplify the rational functions.
Both vertical asymptotes and holes are places that the curve can't quite seem to touch. Holes occur at places where the limit of the function exists, but the function itself does not. For rational functions, holes correspond to the roots (or zeros) of the denominator that cancel out entirely during simplification. Vertical asymptotes occur at places where the limit of the function is ∞ or -∞, which happen at the roots of the denominator that are left over after simplification. It's an important distinction.