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Let's take a look at the limit as x goes to negative two of x divided by the quantity
x squared minus 4 squared. First of all, what happens when we plug in negative 2? On the
top, we get negative 2, on the bottom we get negative 2 squared, which is 4, minus 4, that
gives us zero. So, I've got the form negative 2 over zero -- that tells me that I'm at a
vertical asymptote. So the form of a non-zero number over zero always tells us that we're
at a vertical asymptote. Now we know that there is a vertical asymptote at x equals
negative 2. Knowing this tells us a lot about the limit as x goes to 2. It tells us that
one either side of this asymptote, the function either going to negative infinity or positive
infinity. Therefore, we need to check both sides of that asymptote to see what's actually
happening. To do this, we are going to simply look at the one sided limits, as x goes to
negative 2 from the right and left. Let's start with the limit as x goes to negative
2 from the right of x over x squared minus 4 quantity squared. Because we are approaching
a vertical asymptote, we know that this answer has to be either positive infinity or negative
infinity. We simply need to check the sign of the function really close to negative 2
to see which of those is the case. The way we usually do this is by checking the sign
of the numerator and denominator. We are going to negative 2 from the right, so that's values
that are just a little bit greater than negative 2. Therefore our numerator will be negative
because a value that is just a little bit greater than negative 2 is still negative.
The denominator is going to be positive because the quantity is squared, it will always be
positive. We have a negative over a positive, so that's a negative number overall. So the
limit as x goes to negative 2 from the right is negative infinity. We also need to check
the limit as x goes to negative 2 from the left; once again the possibilities are only
positive and negative infinity, and we're going to also do a little sign chart. Negative
2 from the left, that's values slightly less than negative 2. Once again my numerator is
going to be negative, and once again our denominator is going to be positive since it is squared.
This is negative overall again, and I see that I'm approaching negative infinity as
I go to negative 2 from the left. Because my one sided limits are the same -- they're
both negative infinity - that allows me to say our final answer of the limit as x goes
to negative of this function equals negative infinity. We can also always check our answer
with a graph. This is the graph right here, and we are going to negative 2 remember, so
that is this part right here, and you can see that we are going to negative infinity.