Determine \begin{equation}\lim_{x\to 5}\,\frac{2x^3-5x^2+6x-15}{x-5}\end{equation}

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Lesson Parent: 
Function Limits III: The Gory Details
Problem: 

Determine \begin{equation}\lim_{x\to 5}\,\frac{2x^3-5x^2+6x-15}{x-5}\end{equation}

Answer: 

The limit does not exist.

SAMPLE SOLUTION

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Solution: 

The first thing to try is to find $f(5)$. We immediately see that there is a problem here. The denominator is $x-5$ and as $x$ approaches 5, <Sign in to see all the formulas> has a zero in the denominator. This function has a hole in it at $x=5$ and is actually undefined there. The function does not exist at $x=5$! So can we find the limit?

Recall that we never actually get to $x=a$ because we restrict $x$ to <Sign in to see all the formulas> in the proof of

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In this particular problem, $a=5$ and therefore we are never actually at $x=5$. Because of this we can attempt to divide the numerator by $x-5$ as we did in this problem. Unfortunately, we cannot factor $x-5$ out of the numerator:

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In fact,

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is of the form

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where

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and

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But we know that in this case

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does not exist.