Function Limits IX: Horizontal Asymptotes And The Limit Of $f(x)$ As $x$ Tends To Infinity
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We are assuming that you are comfortable with the gory details of proving function limits so we will just jump right into the definition:
Definition of <Sign in to see all the formulas> |
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For every <Sign in to see all the formulas> there exists a number <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then<Sign in to see all the formulas> |
Definitions like this can be confusing. The point is that we need to find a $Y$ for every <Sign in to see all the formulas>. There is no method for finding this. You can make a guess for it, but usually a mathematician will do some algebra to find a relationship between <Sign in to see all the formulas> and <Sign in to see all the formulas> to determine an appropriate guess. In this lesson we call this guess <Sign in to see all the formulas>. Once a guess is made then the definition above means the following:
| Step | Description |
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| 1 | Make an educated guess for <Sign in to see all the formulas>. |
| 2 | Pick some positive number for <Sign in to see all the formulas>. Since we need to do the proof for all numbers we just "pretend" that <Sign in to see all the formulas> is a number and continue to denote "that number" by <Sign in to see all the formulas>. |
| 3 | Now assume that we are at some value of $x$ such that <Sign in to see all the formulas>. |
| 4 | Now plug <Sign in to see all the formulas> into <Sign in to see all the formulas>. |
| 5 | Now do some basic algebra to see if you can get <Sign in to see all the formulas> into the form <Sign in to see all the formulas>. If you can then you have made a correct guess for <Sign in to see all the formulas> and the proof is complete. If you cannot get it into the form <Sign in to see all the formulas> then there are a couple of reasons:
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The best way to get comfortable with these steps is to try several problems. The easiest one and the one you should keep in your mind is<Sign in to see all the formulas>
Remember that the steps above do not tell us what $L$ is. Once we have a "guess" for $L$ then the definition of a limit proves the assertion
<Sign in to see all the formulas>
In most beginning Calculus courses you usually don't have to worry about finding $L$. You will be given $f(x)$ and $L$ and you will then need to go through the steps above to prove that
<Sign in to see all the formulas>
When this limit exists we say that the function has a horizontal asymptote at $L$.
We can also define <Sign in to see all the formulas>:
Definition of <Sign in to see all the formulas> |
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For every <Sign in to see all the formulas> there exists a number <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then<Sign in to see all the formulas> |
It is possible for a function to have two horizontal asymptotes: one for when $x$ goes to positive infinity and the other when $x$ goes to negative infinity. We would say that
<Sign in to see all the formulas>
and
