Example of one sides unbounded limits

2m read

·Nov 11, 2024

We're asked to select the correct description. It looks like all the descriptions deal with what is the limit of f of x as we approach six from either the right-hand side or from the left-hand side.

So let's think about that. First, let me just do the left-hand side. The limit of f of x as we approach six from the left-hand side: what is this going to be equal to?

As we approach from the left-hand side, we can see, well, f of four is a little under two. f of five is looks like it's around three. f of 5.5 is looks like it's a lot higher. f of 5.75 even, it's just going off the chart. So it looks like this is going unbounded in the positive direction.

So we could say that this right over here is positive infinity. If we were to think about the limit of f of x as we approach 6 from the right side, what is this going to be?

Well, here f of seven, it's negative. f of 7.5 is even more negative. f of 6.5 is even more negative. Not 7. f of 6.5 is even more negative. f of 6.1 is way more negative than that. f of 6.01 would be even more negative than that.

So it looks like this is unbounded in the negative direction. So this is negative infinity.

Now let's see which of our choices match up to that. The way they listed it, they listed the limit from the right direction first. So that's this one: the limit of f of x as x approaches 6 from values greater than six, we have negative infinity.

So that is these two choices; these two choices say that. So we're going to rule out those two choices. Then we think about as we approach six from the left-hand side. We see that we go to positive infinity and that is this choice right over here.

So we rule out that one, and that is what we will pick.

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Example of one sides unbounded limits

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