The graph of f in the figure has vertical asymptotes at x=1 an...

Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits.

lim x→2 f (x)

Accepted Answer

Welcome back, everyone. Analyze limit of FFX as X approaches 6 using the graph of I shown. The dashed lines represent the vertical asymptotes at X equals 3 and X equals 6. So basically we want to evaluate the limit. As x approaches 6 of F of X, and because X equals 6 is a vertical asymptote, that limit might be equal to positive or negative infinity, or it might not necessarily exist, right? It will depend on the limit existence theorem. So we want to consider two limits. The first one is going to be limit as X approaches 6 from the left of F of X. And when we approach 6 from the left we can see that the closer we get to 6 from the left, the more our curve approaches and infinitely. Large negative number. Now, our limit is therefore equal to negative infinity. Now let's evaluate the limit as X approaches 6 from the right. Of F of X. When we approach 6 from the right. The closer we get to 6 from the right, the more our curve extends to positive infinity. So the limit is going to be positive infinity. And based on the limit existence theorem, if these two limits are not equal to each other, which is the case, then this means that the limit as X approaches 6 itself does not exist. Because the two limits must be equal to each other for the limit to exist. And that's our final answer. The limit does not exist. Thank you for watching.

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→2 f (x)

Key Concepts

Vertical Asymptotes

Limits

One-Sided Limits

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