Task 2: Consider the graph of f (Figure 2) and the given...

Task 2: Consider the graph of \(f\) (Figure 2) and the given limits to sketch the graph of \(g\) (Figure 2). Figure 2: Graph of \(f(x)\) and \(g(x)\) Graph of \(f\) Graph of \(g\) 2 2 1 1 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -1 -1 -2 -2 1. \(\lim_{x \to 0} (f(x) + g(x)) = 0\) \(g(x)\) \(g(x)\) \(g(x)\) 2. \(\lim_{x \to 2^-} \frac{\ }{f(x)} = \lim_{x \to 2^+} \frac{\ }{f(x)} = \lim_{x \to 2} \frac{\ }{f(x)} = 0\) 3. \(\lim_{x \to 1^+} (f(x) + g(x)) = 0\) \(f(x)\) 4. \(\lim_{x \to 1^+} \frac{\ }{g(x)} = -1\) 5. \(\lim_{x \to 3^+} (3f(x)g(x)) = -6\) 6. \(\lim_{x \to 3^-} f(g(x)) = 2\)

Transcript

00:01 So in this problem, we're given these two graphs of f and g, for f of x and g of x here.

00:10 Then we're asked a series of limit questions, and we're asked to explain if the limit does not exist, why it does not exist.

00:21 Okay.

00:22 So the first question is the limit as x approaches two of f of v of v.

00:35 Plus g of x well by limit laws the limit of the sum of two functions is the sum of the limits so that means this is the limit as x approaches to of f of x plus limit as x approaches two is supposed to be two right two of g of x well the limit as x approaches two of f of x means i come in here from the left and here from the right.

01:15 And so that is both zero, right? because the function value is zero coming in there from left and right.

01:24 So this is zero plus the limit going into two is x approaches two of g of x.

01:35 Well, that's going this way and this way on that curve.

01:41 Okay.

01:42 And so again, that's going to be zero, isn't it? i'm black.

01:51 So that is zero is the first answer.

01:56 Okay.

01:58 B says the limit as x approaches zero of f of x plus g of x.

02:10 And again, by limit laws, that means this is the limit as x approaches zero of f of x plus.

02:19 The limit as x approaches 0 of g of x.

02:24 Okay.

02:26 So the limit as f approaches zero.

02:30 So coming from the left, that means i'm coming down here.

02:33 And from the right means i'm coming down here.

02:35 And those two are not the same.

02:39 So this does not exist as the limit as x approaches 0 of f of x does not exist...

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