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DM
Numerade Educator

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Problem 7 Easy Difficulty

For the function $ g $ whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $ \displaystyle \lim_{t\to 0^-}g(t) $
(b) $ \displaystyle \lim_{t\to 0^+}g(t) $
(c) $ \displaystyle \lim_{t\to 0}g(t) $
(d) $ \displaystyle \lim_{t\to 2^-}g(t) $
(e) $ \displaystyle \lim_{t\to 2^+}g(t) $
(f) $ \displaystyle \lim_{t\to 2}g(t) $
(g) $ g(2) $
(h) $ \displaystyle \lim_{t\to 4}g(t) $

Answer

(a) $\lim _{t \rightarrow 0^{-}} g(t)=-1$
(b) $\lim _{t \rightarrow 0^{+}} g(t)=-2$
(c) $\lim _{t \rightarrow 0} g(t)$ does not exist because the limits in part (a) and part (b) are not equal.
(d) $\lim _{t \rightarrow 2^{-}} g(t)=2$
(e) $\lim _{t \rightarrow 2^{+}} g(t)=0$
(f) $\lim _{t \rightarrow 2} g(t)$ does not exist because the limits in part (d) and part (e) are not equal.
(g) $g(2)=1$
(h) $\lim _{t \rightarrow 4} g(t)=3$

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Video Transcript

in this problem were given this graph of G of T. So this is the graph of T. I feel like this. All right. T uh t Okay. And were asked a series of limit questions here. The first four asked, is the limit as T goes to zero from the left. That's what full minus their means of our function. So as I'm approaching zero from the left of zero is 40 a zero. Is that coming down this curve right here? And that runs into what value of our function? Well Runs into -1, doesn't it? Okay Then, we're asked to limit as T approaches zero from the right of our function. Okay, So that means I'm coming down this curve now from the right, aren't I? And that goes down to -2, doesn't it? It's what it approaches. Okay then we are asked they were asked Limit as T approaches zero of G. F. T. Well, since the limit from the left and limit from the right are not equal then this does not exist because for that to exist. The limit from the left and the limit from the right, both have to be the same. All right, next, we're asked for the limit as T approaches two from the left. G. O. T. Okay, approaching two from the left, that means I'm going up this curve, aren't I? Coming into two T of it equal to two? But from the left and that goes up to a value of two doesn't Okay. Then we're asked the limit as T approaches to from the right oh G. O. T. Of our function. Well, coming into two from the right means I'm coming down this curve and I'm getting closer and closer to zero, aren't I? Functional Value of 0? Okay. And then f were asked the limit as T approaches to of GFT well again from the left and from the right, these are not equal and so this does not exist, does it? Okay. Next were asked uh Extras to determine G. 2? No, Do you two would be that point right there, wouldn't it? Uh huh. One then. Okay. And then we're asked the limit as T approaches for G. O. T. So here's four. All right. If I go up from four, I'm there. Okay. And if I look at the diagram then This goes across at three here, doesn't it? Okay. And if I'm coming in from the left I get to that point right there, coming in from the right, I get to the same point. So this is three because I'm getting both left and right limits are three as I approach for