The graph of ℎ in the figure has vertical asymptotes at x=−2 a...

Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits.

lim x→3 h(x)

Accepted Answer

Welcome back, everyone. In this problem, we want to analyze the limit of G of X as X approaches to using the graph of G shown. The dash lines represent the vertical asymptotes at X equals -3 and X equals 2. Here we have the graph of G and for our answer choices A says it's negative infinity, B infinity, C 0, and D says the limit does not exist. Now, how can we find the limit of geoX as X approaches too? Well, when we visually inspect our graph, OK, then the limit, OK, the limit as X approaches 2 of GaX exists, OK. If I could write if right beside my term here it exists if. The limit of gess. Let me just clean this up for a minute. The limit. Of geo X as X approaches 2 from the left is equal to the limit as X approaches 2 from the right of geo X. In other words, if the left-hand limit equals the right-hand limit, the limit exists and will be equal to that value. So let's look at GeoEx as it approaches 2 from the right and the left and see if both of those are the same. Now, let's look at it first approaching from the left. Now here is where X E equals 2, and as X approaches 2 from the left, notice that G grows arbitrarily large in magnitude and is as negative, OK, or it continues to get more negative as X approaches 2 from the left. Thus, the limit of G of X as X approaches 2 from the left is going to be negative infinity. Now the question is, will that be equal to the limit from the right as it approach as x approaches to? Well, notice that as X approaches 2 from the right, again G also grows arbitrarily large but in the positive direction. So therefore, the limit of G of X as X approaches 2 from the right will be equal to positive infinity. In this case, we know negative infinity is not the same as positive infinity. So since the left hand limit does not equal the right hand limit, then that tells us that the limit of geo X. As x approaches to does not. Exist If we go back to our answer choices, that means that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→3 h(x)

Key Concepts

Vertical Asymptotes

Limits

One-Sided Limits

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