Determine the limit \[ \lim_{x\to 2}\,\frac{5x^2-3x-14}{x-2}. \]

Primary tabs

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

Determine the limit
\[
\lim_{x\to 2}\,\frac{5x^2-3x-14}{x-2}.
\]

Answer: 

17

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

We can't just determine the limit by sticking the number $2$ in. The reason for this is that there is a zero in the denominator.

Recall our definition of a function limit. It states that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. Our goal here is to find $L$ in

<Sign in to see all the formulas>

It is important to recall from the definition that we do not need to have $x$ actually equal to $2$. We are just getting close to $2$ and not actually reaching. This is seen in the definition where <Sign in to see all the formulas>. However, even if we don't get to $2$ the denominator will get larger and larger as we get closer and closer to $2$.

In these situations we want to see if there is an $x-2$ in the numerator that will cancel the $x-2$ in the denominator. Using the FOIL method you learned in your High School algebra class you can verify that

<Sign in to see all the formulas>

Since $x$ never equals $2$ we can divide the numerator by $x-2$ to get

<Sign in to see all the formulas>


Since we $5x+7$ is a polynomial and we know what the function limit of a polynomial is, we know that

<Sign in to see all the formulas>

The proof is complete, but it is instructional to go through the proof specifically for this problem. Let's choose <Sign in to see all the formulas>. Now let's assume that we are at an $x$ such that <Sign in to see all the formulas>. We now need to show that <Sign in to see all the formulas>. To accomplish this let's do the following algebraic steps:

<Sign in to see all the formulas>


In the one of the above steps we could multiply and divide by $x-2$ because our definition restricts $x$ to <Sign in to see all the formulas>. In other words, because <Sign in to see all the formulas>.