The Form 1^∞

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take one plus one over x, and raise it to the power x. Now let x grow bigger and bigger. The base gets closer and closer to one, and the power runs off to infinity. And one to the power anything is one, so the answer is one, right? Hmm. Put x equals one million into a calculator, and you get two point seven one eight two eight, and then more digits. Not one at all! It is very close to the famous number e, the base of natural logs. So what went wrong? Let us find out. Limits of this type come up often in JEE, and today you will learn a one line rule for them. So, let's begin.

0:46Why 1 to the power infinity is not simply 1

So, first, the symbols. f of x tends to one: the arrow means tends to. And g of x tends to infinity, the sideways eight. Then f of x, to the power g of x, is called the one to the power infinity form. Okay? Now think. The base is close to one, but usually not exactly one. And the power is huge. So is the answer simply one? No, not always! The base is a tiny bit away from one, and it is raised to a huge power. Two effects pull against each other, so we cannot just say one. So we use a trick: f to the power g equals e to the power, g times log f. The l n you see is log, the natural log, to the base e. With me? Now look at log f. Write f as one plus, f minus one, okay? Since f tends to one, f minus one is tiny. And log of one plus a tiny number is almost that tiny number. The wavy equals sign means approximately equal. So log f is approximately f minus one. Clear? Put that in the power, and we get the rule. The short form lim means limit. The limit of f of x to the power g of x equals e to the power, the limit of g of x times, f of x minus one. In words: power times, base minus one, goes up into the power of e. That's the whole trick! But use it only when the base tends to one and the power tends to infinity. Always check the form first, right?

2:20Two quick shortcuts

Now, two shortcuts straight from the rule. First, as x tends to infinity: one plus a over x, in brackets, to the power b x, where a and b are constants. Base minus one is a over x, and the power is b x. So, what is power times, base minus one? Yes, exactly, a times b! The dot means times, so b x times a over x is just a b; the x cancels out. So the limit is e to the power a b. Very good! The second shortcut is the same idea, as x tends to zero. One plus a x, to the power b over x. Base minus one is a x, the power is b over x, and the product is again a times b. So this also tends to e to the power a b. Got it? Now, remember our puzzle? One plus one over x, to the power x. That is a equals one, and b equals one. So the limit is e to the power one, which is just e, about two point seven one eight. That is exactly what the calculator showed. Mystery solved!

3:31Other forms: 0 to the power 0 and infinity to the power 0

One more thing from the course. What if the form is zero to the power zero, or infinity to the power zero? Then there is no base minus one to use, so we take logarithms. The limit of f to the power g equals e to the power, the limit of g times log f. Okay? Look at this course example: the limit of x to the power x, as x tends to zero from the right. The small plus sign on the zero means from the right. Which form is this? Zero to the power zero, correct! The base and the power both tend to zero. So we write x to the power x as e to the power, x log x. Clear? Now, x log x tends to zero. You will see why in section two point three. So the limit is e to the power zero, which is one. Zero to the power zero gave one here, but never assume it; always take logarithms and work it out. With me?

4:31Example 1: limit of (1 + 3/x)^(2x) as x tends to infinity

Now, Example one, an easy one from the course. Find the limit, as x tends to infinity, of one plus three over x, in brackets, to the power two x. Pause the video, and try it first. Done? Okay, let's check. First, the form. The base, one plus three over x, tends to one. The power, two x, tends to infinity. So it is one to the power infinity, and the rule applies. Got it? Now, many students remember that one plus a over x, to the power x, gives e to the power a. Here a is three, so the answer is e to the power three, right? Oh no, careful! The power here is two x, not x. Power times, base minus one: two x times three over x. The x cancels, and we get six. So the limit is e to the power six. Perfect!

5:30Example 2: limit of ((x + 2)/(x - 1))^x at infinity

Now Example two, a little harder, also from the course. Find the limit, as x tends to infinity, of x plus two over x minus one, in brackets, to the power x. Pause the video and try. Hint: check the form first. Done? Okay, let's check. The base, x plus two over x minus one: top and bottom have equal degrees, so it tends to one over one, that is one. And the power x tends to infinity. So, one to the power infinity again! Okay? Now, base minus one. Over the common denominator x minus one, the top is x plus two, minus the bracket x minus one. x minus x is zero, and two minus one is one. So the top is one, right? No, no, wait! The minus sign hits both terms in the bracket. Minus, x minus one, is minus x, plus one. So the top is two plus one, which is three. Base minus one is three over x minus one. This sign slip is very common, so watch it! Clear? Now multiply by the power x. The cross means times. We get three x over x minus one. Equal degrees again, so it tends to three over one, which is three. With me? So the limit is e to the power three. Very good! See, the base did not look like one plus a over x at all, but power times, base minus one, handled it easily. That's it!

7:03Quick revision: the e rule, e^(ab) shortcuts, 0^0 forms

So, quick revision. One: for the one to the power infinity form, the limit is e to the power, the limit of power times, base minus one. But only when the base tends to one and the power tends to infinity. Two: the shortcuts. One plus a over x, to the power b x, and one plus a x, to the power b over x, both give e to the power a b. Three: for zero to the power zero, and infinity to the power zero, take logarithms. Now go and try the practice problems on this page. All the best!