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Calculus/Limit Question

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Expert: - 4/7/2009


Question
lim    1/x - 1/4
x->4   ---------
         x-4


When I do this, I get the denominator of zero, which makes it undefined... However, it is not a solution. So what do I need to do in this problem?? I'm really confused because I run into a lot of situations where the denominator is zero, making it undefined, but the answer never is that option. How do I know when it doesn't exist, versus it existing when i get an undefined statement??

Answer
Multiply the top and bottom by 4x.
On top the result is 4 - x.  On bottom, leave the result as 4x(x-4).
This will give you (4-x)/[4x(x-4)].

Now note that the 4-x cancels with the x-4, leaving a -1/4x.

Any time there is a x-a and the limit is as x->a, try and cancel an x-a somewhere.  When you have complex fractions like this, it is best if you convert them into simpler fractions as was just done.

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