Problem 4.(16 points) Find the limits_ if they exist, or type DNE for any which do not exist:Ixlim (xJ)-(QO) 4r" + 4y21) Along the x-axisz2) Along the y-axis:3...

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{4-x}$$

We're looking at the limit of a polynomial function divided by another polynomial function. So what we could do is we could try direct substitution or substituting substituting in for for why? And then we would get four minus four divided by four squared minus 16, which we can see would be equal to zero divided by zero. And so that's not gonna work. We can't just directly substitute in for for why? So what we're going to do is we're actually going to factor the denominator since it's a difference of squares, we're allowed to factor it into why plus Squared of 16, which is four and then why minus a squared of 16, which is four. And we can see that we have this y minus four factors in the numerator as well as this one in the denominator that will be able to cancel out. And so this is going to be equal to the limit as why goes to four of one divided by y plus four. And now we can directly substitute and four for why? And we get this is one divided by four plus four, Which is equal to one divided by eight.

So if we try to just plug in zero for why we can see that we're going to get zero and the denominator, since the denominator is just equal to why? So we don't want to do that. Um What we're gonna want to do is actually multiply these factors together. So four minus Y times four minus Y. And then we're going to go from there, so four minus Y, times four minus Y. is 16 -8 y plus Y squared. And then we have -16 and divided by why? So you can see here that this positive 16 is going to cancel it. This negative 16 and we're going to be left with the limit As y goes to zero of -8, Y plus Y squared divided by why. And now we're able to actually factor out a factor of why in the numerator. And so this is equal to the limit as why goes to zero of Y times y minus eight and then we have divided by why? So we can now cancel out these two white terms. And now we're just looking at the limit As y goes to zero Of why, -8, and we can just directly plug in zero for why here, And this is going to be equal to -8.

So we're looking at this limit as T. goes to 14 of negative pipe. So we can see that there are no t variables in this limit that we're looking at, We're just looking at the constant negative pi. So since there is no T. Values here um we can assume that if we plug in t. It's not going to actually affect our function at all. Um Since there is no T variable here. So what we can say is that as T. Goes to 14, this is still equal, it's a negative pi. And the reason for that is because if we let fft equal negative pi and we looked at this function um on a graph and we wouldn't have to look at it on a graph but just easier to visualize it. And we went down to a point at negative pi say that's negative pi and we drew a straight line here. Uh huh. Yeah okay there we go. Uh We can see that for any value of T. If this was the T. Axis and this is the FFT axis for any value of T. We're always gonna be at F. F. T. Is equal to negative pi. So when he goes to 14 were still at negative pi. And that's why the limit as T. Goes to 14 is equal to negative pi.

So when we're doing a limit like this, before we go ahead and just use direct um direct substitution are just plugging in this point here. Um What we wanna do is make sure that our function in this case F of X is just equal to X. Um We want to make sure that it has no restrictions on its domain and especially at the point that we're looking at and that it's not going to give us um zero in the denominator or um something something weird like that, or a negative number in under square root sign or something that's not in our domain. In this case the domain of the function F of X is equal to X is all real numbers. So negative four is an all real number. Um So or is a real number? Um Therefore we can just directly substitute negative four in for X. And we can say that this is going to be equal to negative four. Um And the reason for that is because again, when we try to directly substitute this in, we don't have any um things that are going wrong, or domain does include this point at negative for. Um So that's why we can just directly substitute it in this case and say that the limit is exposed negative four of the function F of X Is going to be equal to -4.