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

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>.