Limits at Infinity

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take the square root of x squared plus x, and subtract x. Now let x grow bigger and bigger, without end. Both parts run off to infinity. So infinity minus infinity, that should be zero, right? Hmm. Put x equals one million into a calculator, and you get zero point four nine nine nine nine nine, and then more digits. Almost exactly one half! Not zero. Let us see why. Questions on limits at infinity come up often in JEE, so let's begin.

0:39Divide by the highest power in the denominator

So, first, the symbols. The sideways eight is infinity, and the arrow means tends to. So x tends to plus or minus infinity means x grows without end, either way, okay? Now the one fact behind this whole topic. Any constant c, over x to the power k, with k greater than zero, tends to zero. Divide a fixed number by something enormous, and you get almost nothing. Right? So here is the method. Divide the top and the bottom by the highest power of x in the denominator. After that, every term is either a plain number, or something that tends to zero. Clear? Let us try one from the course. x plus one, over x squared plus one, as x tends to infinity. Which power of x do we divide by? x squared, yes! The highest power in the denominator. The top becomes one over x, plus one over x squared. The bottom becomes one, plus one over x squared. Now each small term tends to zero: zero plus zero, over one plus zero. The limit is zero. Got it?

1:51Compare the degrees

Now, the whole picture. On top, a polynomial of degree m: a naught, that is a with a small zero below it, times x to the power m, and the three dots mean the lower powers. At the bottom, degree n, with leading coefficient b naught. The equals sign with a slash through it means not equal to: the leading coefficients are not zero. Okay? If the top has the smaller degree, m less than n, the bottom wins, and the limit is zero. That is exactly what we just saw. With me? Now, equal degrees. We divide by x to the power n. Which terms survive? Only the leading terms, exactly! Everything else tends to zero. So the limit is a naught over b naught, the ratio of the leading coefficients. Clear? And last, the top has the bigger degree, m greater than n. Now the top wins, and the expression grows without bound. We write plus or minus infinity; the sign depends on the leading coefficients and the direction of x. Okay? Like this course example: two x cubed plus one, over x squared plus five. Degree three on top, degree two below. So as x tends to infinity, it tends to infinity. That's it!

3:13Careful with square roots as x tends to minus infinity

Now, a sign trap that catches many students. Look at the root sign over x squared. The square root of x squared is mod x: the two vertical bars mean the absolute value. Not just x! And when x is negative, mod x equals minus x. Remember, a square root is never negative. Okay? Try this course example. The short form lim, for limit, with x tends to minus infinity written below it: root of x squared plus one, over x. Root of x squared is x, so it is x over x, and the answer is one, right? No, no, careful! Take x squared out of the root, and it comes out as mod x. Here x is negative, so mod x is minus x. So root of x squared plus one equals minus x, times root of one plus one over x squared. Clear? Now divide by x. The x cancels, leaving minus root of one plus one over x squared, which tends to minus one. Minus one, not one! Quick check: the top is positive and x is negative, so the fraction stays negative, and the answer can never be plus one. Got it?

4:28Powers: divide by the largest base

One more tool, for powers, where n is a natural number tending to infinity. If mod a is less than one, a to the power n tends to zero. If a is greater than one, a to the power n tends to infinity. With me? Now this course example: two to the power n, plus three to the power n, over three to the power, n plus one, minus two to the power n. Which power should we divide by? The largest base: three to the power n, yes! Two to the power n over three to the power n becomes two over three, in brackets, to the power n, and that tends to zero. Three to the power n over itself is one, and three to the power, n plus one, over three to the power n is just three. So we get zero plus one, over three minus zero: the limit is one over three. Perfect!

5:22Example 1: limit of (3x2 + 5)/(2x2 - x) at infinity

Now, Example one, an easy one from the course. Find the limit, as x tends to infinity, of three x squared plus five, over two x squared minus x. Pause the video, and try it first. Done? Okay, let's check. The top tends to infinity, and the bottom tends to infinity. Infinity over infinity is one, right? Oh no, careful! Infinity over infinity does not give the answer; it only tells us to do more work. So divide top and bottom by x squared, the highest power in the denominator. The top becomes three plus five over x squared, and the bottom becomes two minus one over x. Okay? Now the small terms tend to zero: three plus zero, over two minus zero. Three over two. The shortcut agrees: equal degrees, so the ratio of the leading coefficients. Very good!

6:20Example 2: root(x2 + x) - x at plus and minus infinity

Now, remember our puzzle from the start? Let's finish it, Example two from the course. Find the limit of root of x squared plus x, minus x, first as x tends to plus infinity, and then as x tends to minus infinity. Pause the video and try. Hint: rationalise. Done? Okay, let's check. As x tends to plus infinity, this is infinity minus infinity, so we rationalise. Multiply and divide by root of x squared plus x, plus x. The top becomes x squared plus x, minus x squared: just x. So we get x over, root of x squared plus x, plus x. Okay? Big over big again. What do we divide by now? By x, yes! For positive x, root of x squared is just x, so x comes out of the root cleanly. We get one over, root of one plus one over x, plus one. That tends to one over one plus one: one half! There is our puzzle answer. Infinity minus infinity is not automatically zero. Got it? Now x tends to minus infinity. Same expression, so one half again, right? No, no, wait! Look at each part. Root of x squared plus x is positive, and it tends to plus infinity. And minus x? x is hugely negative, so minus x tends to plus infinity too. With me? Two huge positive numbers added: the sum tends to plus infinity. No infinity minus infinity form here, so nothing to rationalise. Clear? So, two directions, two different answers: one half, and plus infinity. x tends to plus infinity and x tends to minus infinity are separate limits, so always check the direction. Remember that!

8:20Quick revision: degrees, rationalising, largest base

So, quick revision. One: for a polynomial over a polynomial, divide by the highest power in the denominator, then compare the degrees. Two: infinity minus infinity with square roots, rationalise; and as x tends to minus infinity, root of x squared is minus x. Three: for powers, divide by the largest base. Now go and try the practice problems on this page. All the best!