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Let's take a look at the limit as x goes to 2 of x over the quantity of x minus 2 squared.
Let's try to evaluate this limit together. To start out, let's just go ahead and plug
in 2 and see what happens. If I plug in 2 to the top, I'm just going to get 2 because
it's the limit as x goes to 2 of x. If I plug in 2 to the bottom, I'm going to get 2 minus
2 squared which is going to give me 0. So, I have the form 2 over 0. Now, from what we've
just learned, we know that if we get the form of a number over 0, it tells us that we're
at a vertical asymptote. Now I know that there is a vertical asymptote at x equals 2, because
when I plugged in 2 to the function I got a number over 0. Knowing that there is a vertical
asymptote at x equals 2 tells us a lot about what the limit is going to be as x goes to
2. So, because we know there is a vertical asymptote there, we know that on either side
of x equals 2, the function is going to be going to either positive or negative infinity.
To determine what the limit is overall, we need to look at the one sided limits. Let's
start with the limit as x goes to 2 from the right. So, because I'm approaching a vertical
asymptote, 2, I know that there are only two options for this limit. This limit could be
either positive infinity or negative infinity, and I need to find out which one by checking
the sign of the function just to the right of 2. So what do I mean by that? In other
words, what's the sign of this fraction x over x minus 2 squared for values that are
just a little bit greater than 2. Let's picture plugging in like 2.001. If I plug in 2.001
to this function, is the numerator positive or negative? Well it's positive, 2.001 is
a positive number. What about the denominator? If I plug in 2.001, the bottom is also going
to be positive right? I've got a positive over a positive, that's a positive overall,
so I know that I'm going to positive infinity on this side of 2. Now I need to do the left
sided limit in the same manner. The limit as x goes to 2 from the left of x over x minus
2 squared. So once again there are still only two options. I know I'm approaching a vertical
asymptote and I'm coming on from the left hand side. I just need to figure out if it's
positive or negative infinity and we're going to do it in exactly the same way. Let's think
about the top of the fraction. For values that are just slightly less than 2, the top's
going to be positive. What about the bottom? The bottom is squared so it's always going
to be positive right? If I plug in 1.99 minus 2 and square it, that's a positive number.
Once again, a positive over a positive gives me positive overall. What do I see? I see
that my left handed limit equals positive infinity, and my right handed limit equals
positive infinity so that tells me the answer for my limit overall. The limit as x goes
to 2 of this function equals positive infinity because the one sided limits match up.