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For the function $ g $ whose graph is given, state the following.

(a) $ \displaystyle \lim_{x \to \infty} g(x)$(b) $ \displaystyle \lim_{x \to - \infty} g(x)$(c) $ \displaystyle \lim_{x \to 0} g(x)$(d) $ \displaystyle \lim_{x \to 2^-} g(x)$(e) $ \displaystyle \lim_{x \to 2^+} g(x)$(f) The equations of the asymptotes

a.2b.$-1$c.$[-\infty]$d.$-\infty$e.$=\sqrt{\infty}$f. y=2 x=0 y= -1 x=2

03:19

Daniel J.

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Baylor University

University of Michigan - Ann Arbor

Boston College

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mhm. In this problem we are given the graph of the function G fx function is given in pink and we need to determine some limits from the graph. Okay. No, the first limit has the limit as X tends to infinity of G fx. Now for that we'll look at the behavior of the graph when X increases in the positive direction. Now note from the graph, in fact, as X increases and it tends towards infinity G fx it becomes closer and closer and closer and it basically approaches the line. Why is equal to 2? Yeah. As X tends towards infinity G F X against words too. So the answer to our first limit equals two. Our second limit, it's the limit of cfx has X approaches or tends towards negative infinity. Now we can use similar reasoning and see that has X decreases without pounds. Yeah, G fx starts to tend words. Mhm Why equals negative Dude, As you can see from this graph. Okay, G Fx approaches negative too. So the answer to our second limit is negative two. Our third limit Is the limit as X approaches zero rather Our 3rd limited to limit of GFX has expert to zero. So noted as X approaches zero from the left hand side. So we're just following the graph. So G Fx starts to decrease without bound. So it decreases decreases, decreases and it tends towards negative infinity. The same is true for the right hand side. An apology effects and it's decreasing without found and approaching negative infinity. Now at the point X equals zero. Both the left and right hand side limits are equal and they're equal to negative infinity. So our limit turns out to be negative infinity. For the next limit you need to find the limit as X approaches to from the positive side or the right hand side of G FX. So as X approaches two from the right hand side, you can see as X approaches to. Yeah, yeah, from the right hand side. Right this okay. Mhm. The value of G is about .7 and the same is true and it approaches to from the left hand side. That means the limits as X approaches to you from the right hand side is about Let me use an approximate symbol years about .7. And as it approaches do from the left hand side. The lemons also .7. And for the last part we need to find the equations of the ASM Dotes. So first well find the ASM does and I'm going to do that using a green Pen. So the first ASM to it is actually why equals two as extends to infinity. That's our horizontal isn't Dote Now there's a cord and other horizontal as in dude at Y equals negative two. Because as as X approaches negative infinity white walls negative two. And as X approaches infinity why approaches to Now there's also a vertical as um Dote at X equals three. Another vertical hasn't Dude Had x equals zero. And those are the equations how far required hasn't dealt with that. We're done with this problem.

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