What is the limit of x3−5x2+3x−15x−5 as x approaches 5?

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Lesson Parent: 
Function Limits I: The Intuition
Problem: 

What is the limit of x3−5x2+3x−15x−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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