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Let's take a look at this limit. The limit as x goes to 4 of 3 minus 2x over x minus
4 times x plus 2. We can start just by plugging in 4 and see what we get. If I plug in 4 to
the top, I'm going to get 3 minus 8 which is negative 5. If I plug in 4 to the bottom,
I'm going to get 0 times 6. That's going to give me 0 on the bottom. I have the form of
a number over 0, which tells me that there's a vertical asymptote here. Now I know that
there's a vertical asymptote at x equals 4. That tells me something about the limit. So,
because I'm approaching a vertical asymptote on either side of x equals 4 I'm either going
to be going to positive or negative infinity. The limit overall could be positive infinity,
negative infinity, or does not exist if the two sides don't match up. We need to go ahead
and evaluate those one sided limits. Let's start with the limit from the right. I want
to think about the limit as x goes to 4 from the right of 3 minus 2x over x minus 4 times
x plus 2. Once again, because I'm at the vertical asymptote, the one sided limit, there's only
two options for what it could be. It could either be positive infinity or negative infinity.
To figure out which one of those is the correct option, I'm going to look at the sign of this
function for values that are just slightly greater than 4. We can just make a little
sort of sign chart to do that. Let's think about the numerator. For values that are just
slightly greater than 4, so maybe like 4.0001, what is the sign of the numerator? That's
going to be 3 minus a number that's really close to 8, so that's going to be negative
on top. What about the bottom? On the bottom we have two factors so let's think about them
separately. What about the x minus 4 factor? For values that are a little bit greater than
4, that's going to be positive. And then the x plus 2 factor, that's also going to be positive
because I have about 4 plus 2 so that's positive. So this is a negative over a positive, which
gives me a negative overall. I know this one sided limit is negative infinity. Let's do
the other side. The limit as x goes to 4 from the left of 3 minus 2x over x minus 4 times
x plus 2. This time I'm approaching 4 from the left hand side. That's values that are
just slightly less than 4. Once again my options are either positive or negative infinity.
Let's create another sign chart. Okay so numerator, just slightly less than 4 so I'm still going
to have 3 minus about 8. That's still going to be a negative on top. What about the bottom?
Let's think about the two factors here. X minus 4, now if I plug in something that's
a little bit less than 4, now that factor's going to be a negative. X plus 2 is still
going to be positive though. I've got a negative over a negative this time, which gives me
a positive overall. I'm approaching positive infinity on this side of the asymptote. What
does this tell me about the value of the limit just at 4 in general? So, the limit as x goes
to 4 of 3 minus 2x over x minus 4 times x plus 2. The one sided limits are different.
On one side the limit is going to negative infinity and on the other side the limit is
going to positive infinity. Then we say that this limit does not exist overall, because
the one sided limits do not match up. Then really quickly let me show you a graph of
this function. This graph here is the graph of the function that we just looked at. We
are considering the limit as x goes to 4, so I'm over here if you see where my mouse
is moving. We see that on one side the limit is going to positive infinity and on the other,
to negative infinity. So, overall the limit does not exist.