Previously, we had 3 ways to evaluate limits: Graphing, a Numeric Approach (using tables), and Direct Substitution.
Today, we added a few more techniques to our ever increasing bag of tricks. After all, a wise man often tells us, "Variety is the spice of life."
When given a problem such as...
You MUST factor to check for holes or asymptotes
From there, you use direct substitution and plug 3 in for x. Therefore, x=-1.
Next, we moved on to another type of situation. When given a problem such as...
You again have to factor this to check for holes/asymptotes. You may be scratching your heads and saying, "How do I factor a cubic?" Well my friends, I'll remind you of something we learned all the way back in Algebra 2. SYNTHETIC DIVISION!!!
To start, you plug the number that x is approaching into your box. In this specific case, we will use 2.
If it factors, there is a hole in the graph and it has a limit. However, if it does not factor, the graph has an asymptote and therefore a limit DOES NOT EXIST.
When you use synthetic division on the original problem,
You get a remainder of -2 meaning that it has an asymptote and therefore not a limit.
Rationalizing the Numerator
In order to solve a problem like this one,
You have to multiply it by the conjugate. Usually, you would do this to rationalize the denominator, but for limit problems you rationalize either the numerator or denominator to help you solve the problem.
This leaves you with...
From there, the x's cancel out and you are left with...
You then use direct substitution and plug 0 in for x. The problem then simplifies to 1/2.
Now for your viewing pleasure, here is a song about limits. Aspects of it hold true to my life. It may for some of you as well. Enjoy!!! :)
Here is the link (the video player wasn't working)
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