Use the graphs of f and g in the accompanying figure to find...

Use the graphs of $f$ and $g$ in the accompanying figure to find the limits that exist. If the limit does not exist, explain why. (a) $\lim _{x \rightarrow 2}[f(x)+g(x)]$ (b) $\lim _{x \rightarrow 0}[f(x)+g(x)]$ (c) $\lim _{x \rightarrow 0^{+}}[f(x)+g(x)]$ (d) $\lim _{x \rightarrow 0^{-}}[f(x)+g(x)]$ (e) $\lim _{x \rightarrow 2} \frac{f(x)}{1+g(x)}$ (f) $\lim _{x \rightarrow 2} \frac{1+g(x)}{f(x)}$ (g) $\lim _{x \rightarrow 0^{+}} \sqrt{f(x)}$ (h) $\lim _{x \rightarrow 0^{-}} \sqrt{f(x)}$ (GRAPH CAN'T COPY).

Transcript

00:01 We're given two functions f, which is the one on the right, or the one on the left, and g, which is the one on the right.

00:08 And we want to find the limit, some limits.

00:13 Okay, so the first thing you need to know is that when you take the limit of an absolute value, it's the same as taking the absolute value of the limit.

00:21 So you can put the limit inside.

00:24 And not only that, but you can put the limit with each thing because the limit of the sums is the sum of the limit.

00:30 Limits okay so you can do like that all right so now we're going to take the limit as x approaches 2 of f of x so here's x equals 2 right here so we're going to approach 2 from the left and we're going to approach 2 from the right and you can see the arrows match up at y equals 0 so that limit is 0 plus same thing on g we're going to here's 2 right here we're approach two from the left we're going to approach two from the right the arrows meet up at y equals zero so the limit is zero zero plus zero zero absolute value zero all right next one we're going to take the limit is x approach of the same function okay so we're going to come at zero from the left on f and we get positive one and we're going to come at zero from the right and we get negative two and since we got two different things, since the arrows didn't meet up, that means the limit doesn't exist.

01:43 So since the limit as x approaches zero of f doesn't exist, then this whole limit doesn't exist.

01:49 We don't even care what the limit is for g.

01:52 So it does not exist.

01:57 Okay, so now what we're going to do is we're going to take the limit as x approaches zero from only the right -hand side.

02:04 Okay, so we're on the right -hand function, which is the right -hand piece of the function, which is the purple piece.

02:12 So we're coming at x equals 0 and we get y equals 2.

02:19 And then on g we're coming at x equals 0 on the blue part and we get y equals 2.

02:28 Oh this one should be y equals negative 2.

02:31 Sorry about that.

02:32 So we have negative 2 plus 2 which is 0.

02:38 So the limit from the right and the left each always exist but the plan limit as x approaches zero doesn't always exist if we don't get the same thing from the left and the right.

02:51 Okay, now we're taking the limit from the left, so we're coming at zero from the left hand side.

02:57 So for f, we're coming on the green part of the function, and even though there's a hole there, the limit is still equal to one, right? we don't care what the function equals to, we just want to know what the graph is getting close to.

03:12 On the g function, we're on the red part.

03:15 We're going towards zero and x goes to two.

03:17 So we get 1 plus 2, which is 3.

03:23 Okay...

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