So that's our x-axis. This right over here is our y-axis. And let's graph f of x. So we see that if x is a very small number, if x is 0. And as the closer and closer we get to 0 from the positive direction, f of x approaches infinity.
So it just keeps approaching infinity as we get closer and closer to 0. As x gets closer and closer to 0, the y value just gets higher and higher. Then, as our x value gets larger and larger, our f of x value gets smaller and smaller. So it looks something like that it approaches 0. Similarly, if we approach x from the negative direction right over here, we saw that f of x is approaching negative infinity. So as we get x is closer and closer to 0, our f of x gets more and more and more negative.
And then as our x becomes more and more negative, the x itself becomes more and more negative. We see that our function is approaching 0.
You have a horizontal asymptote at y is equal to 0. When x approaches infinity, f of x gets closer and closer to 0 but never quite touches it. When x approaches negative infinity, f of x is getting closer and closer to 0 from the bottom but it never quite touches it. And we also have a vertical asymptote right over here at x is equal to 0.
And we see that because as x approaches 0 from the positive direction, y approaches infinity. And as x approaches 0 from the negative direction, y approaches negative infinity. So the limit here, at x is equal to so if you were to say, we looked at the limit as x approaches 0 from the positive direction and from the negative direction, but we see that they're approaching two different things.
So we definitely have a vertical asymptote at x is equal to 0. Then we take a limit once we have an expression which we know how to handle. Then 0 divided by any nonzero number is 0. This one is trickier to understand fully. What I will show is that it cannot approach any particular value and it does not approach the same infinity.
So if both sides were to approach the same number, that number would have to be 0. So in particular it will not approach 0. Thus, the limit does not exist.
So what you're dealing with is called an indeterminate form. The indeterminate forms are strange to look at because there's no immediately clear answer to them when they come up. How do you check that rate of change? You use the derivative AKA the operation that can give you the rate of change of a function. This is a really useful trick called L'Hospital's rule. So it's a cool way to solve some more problems. So when would you put that a limit does not exist?
When the one sided limits do not equal each other. First, a one sided limit is when you approach a value from a single side. You specify which side you are coming from with a plus or minus over what x is approaching.
This is a one sided limit coming from the right side. This is a one sided limit coming from the left. Since the limits from both the right and left are the same. The limit exists and is equal to what both of the one sided limits are equal to. At first glance this may appear to be a contradiction. Upon doing the simplification we can note that,. Also, zero in the numerator usually means that the fraction is zero, unless the denominator is also zero. We might, for instance, get a value of 4 out of this, to pick a number completely at random.
There are many more kinds of indeterminate forms and we will be discussing indeterminate forms at length in the next chapter. However, there is still some simplification that we can do. This limit is going to be a little more work than the previous two. Also note that neither of the two examples will be of any help here, at least initially. When there is a square root in the numerator or denominator we can try to rationalize and see if that helps. Recall that rationalizing makes use of the fact that.
This might help in evaluating the limit.
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