The Idea of a Limit

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Picture a function that is not even defined at x equals two, and yet it has a perfectly good limit there. And another that is defined at x equals one, yet has no limit there at all. Not defined, but a limit? Defined, but no limit? Strange, right? By the end, both will feel completely natural. Continuity and derivatives are built on limits, and questions on them come up often in JEE, so let's begin.

0:34What does a limit mean?

First, the definition. We read this line as: the limit as x tends to a, of f of x, equals capital L, okay? The short word lim stands for limit, and the arrow under it means tends to: x moves closer and closer to a. Capital L is just a number. Clear? Now the key words. Take x close enough to a, but x not equal to a; the equals sign with a slash through it means not equal to. So x comes as near to a as we want, but never lands on a. Right? And then f of x can be made as close to capital L as we like. That is the whole definition. With me? Now, tell me one thing. To find the limit at a, do we need the value f of a? No, not at all! x never equals a, so f of a never enters the picture. The value f of a plays no part. It may be different from L, or it may not exist at all. Okay?

1:41Left-hand and right-hand limits

Now look at this graph, with a jump at x equals one. On the left, the dark blue line climbs to a hollow circle at height two; hollow means that point is not on the graph. On the right, the orange line starts at a filled dot at height three, so f of one is three. The dashed lines mark x equals one and the two heights. Right? Now walk along the blue line towards x equals one, from the left. Which height does f of x come close to? Two, yes! The blue label says it: LHL equals two, the left-hand limit. We write the limit as x tends to a minus; the small raised minus sign after a means from the left. To work it out, put x equals a minus h, with h a small positive number, and let h tend to zero from the right: the zero with a small plus sign. Got it? Now come back along the orange line, from the right: the heights rise up to three. That is the right-hand limit, RHL; the orange label says three, equal to f of one. Here x tends to a plus, with a small raised plus sign, from the right, and we put x equals a plus h. Okay? So, f of one is three, and the right-hand limit is also three. So the limit at one is three, right? No, no, careful! The limit exists if and only if the left-hand and right-hand limits both exist as finite numbers, and are equal. This double-headed arrow means if and only if. One side is never enough. Clear? Here two is not equal to three, so the limit as x tends to one does not exist, even though f of one is three! Our second puzzle: defined, but no limit. With me?

3:30Which functions need both sides?

Some functions need this check almost every time. First, near zero: modulus of x, with two vertical bars; signum of x, written s g n, which gives the sign of x; and modulus of x over x. Okay? Quick check. For positive x, what is modulus of x over x? And for negative x? For positive x, modulus of x is x, so the fraction is one. For negative x, modulus of x is minus x, so it is minus one. LHL is minus one, RHL is one: no agreement, no limit. Right? Next, near an integer. Square brackets mean the greatest integer less than or equal to x; we just say the greatest integer of x. Curly brackets mean the fractional part of x. And piecewise functions have one rule on each side. Now, take x just less than an integer n. What is the greatest integer of x? n minus one, exactly! x has not reached n yet. And just to the right of n, it is n itself. Two different values, so no limit at an integer. Got it?

4:44Two more one-sided limits to remember

Now two more; these come up often. Take e to the power one over x. As x tends to zero from the right, one over x is huge and positive, so e to the power one over x tends to infinity; that sideways eight is the infinity symbol. Clear? Now from the left, where x is a tiny negative number. What happens to e to the power one over x? It tends to zero! One over x is a huge negative number, and e to the power of a huge negative number is almost nothing. From the right, infinity; from the left, zero. Okay? The same story for tan inverse of one over x; the small minus one up there means inverse, not one over tan. From the right it tends to pi over two, and from the left, to minus pi over two. With me?

5:37Example 1: limit of (x squared - 4)/(x - 2) as x tends to 2

Now, Example one: our first puzzle from the start. Find the limit as x tends to two, of x squared minus four, over x minus two; the small two up there means squared. Is the function defined at x equals two? Pause the video, and try it first. Done? Okay, let's check. Put x equals two: four minus four, over two minus two. Zero over zero has no meaning, so the function is not defined at x equals two. Right? So, no limit at two either, correct? No, no, careful! For the limit, x comes close to two but never equals two. And for x not equal to two, x squared minus four is x minus two, times x plus two. Since x minus two is not zero, we cancel it: the fraction is just x plus two. Clear? As x tends to two, x plus two tends to four. So the limit is four, though the function is not defined at two: not defined, but a limit! Puzzle solved. Got it?

6:46Example 2: limit of 1/(1 + e to the 1/x) as x tends to 0

Now a slightly harder one. Discuss the limit as x tends to zero, of one over, one plus e to the power one over x. Pause and try. Hint: check both sides. Done? Okay, let's check. As x tends to zero from the right, e to the power one over x tends to infinity. So the bottom is huge, and one over something huge tends to zero. So RHL is zero. Okay? So the limit is zero, right? Many students write zero and move on. Oh no, careful! We checked only one side. As x tends to zero from the left, e to the power one over x tends to zero. So the fraction tends to one over one plus zero, that is one. LHL is one. Clear? Left gives one, right gives zero. One is not equal to zero, so the limit does not exist. With me?

7:48Quick revision: LHL, RHL and when a limit exists

So, quick revision. The limit at a looks at x near a, never at a, so f of a plays no part. The limit exists only when the left-hand and right-hand limits are equal, and both finite. And for a modulus, a greatest integer, a fractional part, e to the power one over x, tan inverse of one over x, or a piecewise rule, check both sides first. Now go and try the practice problems on this page. All the best!