What is the limit of $\frac{x^3-5x^2+3x-15}{x-5}$ as $x$ approaches 5?

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Lesson Parent: 
Function Limits I: The Intuition
Problem: 

What is the limit of $\frac{x^3-5x^2+3x-15}{x-5}$ as $x$ approaches 5?

Answer: 

28

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?

It is important to note that

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and

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So in this problem we have a situation where both the numerator and denominator are both going to zero. 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 do the division by $x-5$ and rewrite the function as

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to see that

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We know that

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and therefore

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