Method: Vertical Asymptotes

Unit 1: Limits and Continuity Option 9: Vertical Asymptotes (Limits that equal infinity) Method: Vertical Asymptotes

The Big-Small Game

Showing that the result of a limit equation is $\pm \infty$ is a much more conceptual process compared to your other limit options. You cannot calculate infinity, as it has no defined value.

This also means that the work you are able to show is far more limited. Most of the time you will just be asked to identify the vertical asymptotes and give the reason you know. Or you are given a limit problem where the result just happens to be $\pm \infty$ .

Step 1: Decide if you are going to run a left-hand limit, right-hand limit, or overall limit.

If you are just looking to see if you have a vertical asymptote, then you only need the results of one limit (left or right) to be $\pm \infty$ . If you don’t get $\pm \infty$ for one, you will still need to check the other to be certain of your result.

If you are performing this process as a general overall limit problem with no vertical asymptote reference, then you will always need to determine the left-hand limit and the right-hand limit to find the result of an overall limit.

Step 2: Try Option 1: Plug It In

Step 3: Try Option 2: Factor and Cancel

This step helps you to determine which of the parts of a fraction that give you division by zero are actual vertical asymptotes instead of holes (removeable discontinuities) in the graph.

Step 4: Back to Option 1: Plug It In with an $x - value$ that is really, really, really close to the $x - value$ that you actually care about.

You will need to use that $x - value$ to approximate the value on the top of your fraction and the bottom of your fraction.

This is where things start becoming more conceptual and you have to start thinking of numbers in more generic terms like Big and Small . Big numbers are going to be numbers bigger than 1 or less than -1. Small numbers would be decimals that are between 1 and -1. You must also keep track of whether these value or positive ( $+$ ) or negative ( $-$ ).

Just like with any other Option 1: Plug It In limit problem you should have a result for the limit at this point. The result won’t be an actual value it will be conceptual. You will have an idea of what is happening to the top of the fraction, positive/negative Big / Small , and what is happening to the bottom of the fraction, positive/negative Big / Small , but you won’t have an exact value. This is also why you don’t normally need to show work for these steps on a problem. However, you need to find the answer, which means at a minimum you mentally have to go through this process.

Step 5: Use logic to evaluate the final result from Step 3.

The good news is there are only four ways these results can come at you if you are actually looking at a vertical asymptote. Once you have learned these logic options, these should be straightforward to read off.

Result

Concept

Logic

Final Result

$\frac{+ Big}{+ Small}$

Think:

$\frac{+ 1}{+ . 000000001} = + Huge Number$

A positive Big number divided by a positive Small number becomes positive Huge Number.

If you look at the fraction on the left you can see that the closer you get to our $x - value$ the smaller that decimal will become and the larger the final result.

The limit is heading towards $y = + \infty$

You have a vertical asymptote at the limit’s $x - value$ .

$\frac{+ Big}{- Small}$

$\frac{+ 1}{- . 000000001} = - Huge Number$

A positive Big number divided by a negative Small number becomes negative Huge Number.

If you look at the fraction on the left you can see that the closer you get to our $x - value$ the smaller that decimal will become and the larger the final result.

The limit is heading towards $y = - \infty$

You have a vertical asymptote at the limit’s $x - value$ .

$\frac{- Big}{+ Small}$

$\frac{- 1}{+ . 000000001} = - Huge Number$

A negative Big number divided by a positive Small number becomes negative Huge Number.

If you look at the fraction on the left you can see that the closer you get to our $x - value$ the smaller that decimal will become and the larger the final result.

The limit is heading towards $y = - \infty$

You have a vertical asymptote at the limit’s $x - value$ .

$\frac{- Big}{- Small}$

$\frac{- 1}{- . 000000001} = + Huge Number$

A negative Big number divided by a negative Small number becomes positive Huge Number.

If you look at the fraction on the left you can see that the closer you get to our $x - value$ the smaller that decimal will become and the larger the final result.

The limit is heading towards $y = + \infty$

You have a vertical asymptote at the limit’s $x - value$ .

If your Result is not one of the 4 options above, then you have either; (1) Had an error in your conceptual process on Step 3 or (2) you are not looking at a vertical asymptote and another limit option was your better option for solving the problem.

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Unit 1: Limits and Continuity

Method: Vertical Asymptotes

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