What is the difference between infinite limits and limits at infinity

Let us now look at the last two subsections and go deeper. In the last two subsections we looked at horizontal and slant asymptotes. Some increases to infinity faster than others.

Let's state a theorem we mentioned when we discussed the last example in the last subsection:. For now, one can plot and compare the graphs of an exponential function and a power function.

Since the growth rate of a polynomial is the same as that of its leading term, the following is obvious:. Then the following is totally within our expectation:. What about exponential functions with different bases? We use a change of variable. Like this: Like Loading Leave a Reply Cancel reply Enter your comment here Fill in your details below or click an icon to log in:.

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So when x is 0. To 1 over 0. So you see, as x gets closer and closer to 0 from the positive direction, f of x just really grows really, really fast. So what we say here is the limit of f of x as x approaches 0 from the positive direction is going to be equal to positive infinity.

Or we could just write infinity. This thing over here-- if we put something really, really close, so if we say 0 point seven digits behind the decimal place, then 1 over that's going to be 1 with one, two, three, four, five, six, seven zeros. Did I do that right? Here I had four places behind the decimal, four 0's. Here I have one, two, three, four, five, six, seven, and here I have seven 0's.

So you see, as we get closer and closer to 0 from the positive direction, the f of x just gets larger and larger and larger. It's just completely unbounded. So we'd say this is equal to infinity. Well, let's think about another limit. Let's think about the limit as x approaches 0 from the negative direction of f of x, or the limit of f of x as x approaches 0 from the negative direction. Well in that case, we can just make each of these values negative. So if x is negative 0. If this is negative, then this is negative.

And so what we see here is that this gets more and more-- becomes larger and larger numbers in the negative direction. If we keep going, if we're thinking about a number line, further and further and further to the left. So we can say the limit of f of x as x approaches 0 from the negative direction is equal to negative infinity.