]> IB HL 2 - Limits

Limits

Finding Limits Graphically and Numerically

The Idea

It is of interest to mathematicians to know what happens to the value of a function as the input (x) gets really close (arbitrarily close) to a particular value. This is usually because we cannot plug in the specified x-value directly—perhaps because it is not in the domain. This process of determining the behavior of the function near certain x-values is the process of limits.

The Idea, Graphically

Let's look at the function $y = \frac{1}{x - 1}$ around the x-value of 1. Note that 1 is not in the domain of this function.

An Image

What happens to the value of the function (the y-value) as x gets really close to 1? Well, it depends on which side you're looking at. On the left side of 1, the function is headed down. To the right of 1, the function is going up. We call these the left- and right-hand limits, respectively—and we'd say that the limit on the left is negative infinity, the limit on the right is positive infinity. Note that for this function, the left- and right-hand limits are not the same.

Try this function: $y = \frac{x - 2}{x^{2} - 4}$ . Note that neither 2, nor -2, are in the domain.

An Image

What happens as x gets really close to 2? Perhaps we should zoom in on that area.

An Image

As x gets closer to 2 (from the left), the y-values appear to be headed towards 0.25. The same occurs on the right. So the left- and right-hand limits are the same here.

The Idea, Numerically

Let's look at that last one, but from a numeric perspective. Here's a table of values for the function, with x-values very close to (but slightly lower than) 2.

x	1.9	1.99	1.999	1.9999	1.99999
y	0.25641	0.250627	0.250063	0.250006	0.250001

The values of x are getting closer and closer to 2, and the y-values appear to be getting closer and closer to 0.25. The same is true on the right:

x	2.00001	2.0001	2.001	2.01	2.1
y	0.249999	0.249994	0.249938	0.249377	0.243902

The Limit of a Function

First, notation: to talk about the limit of the function f(x) as x approaches the value a from the left, we write $lim_{x \to a^{-}} f (x)$ . The same limit, from the right, is written $lim_{x \to a^{+}} f (x)$ .

Now we can talk about the limit at a value. $lim_{x \to a} f (x)$ exists if $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x)$ . In other words, if the left- and right-hand limits agree, then we can just talk about the limit, without any qualifying words.

There is a technical definition for limits. It's not part of the IB Programme, but let's take look anyway.

The limit of f(x) as x approaches a is L ( $lim_{x \to a} f (x) = L$ ) means that for every positive real value ε, there exists some positive real value δ, so that if $| x - a | < δ$ , then $| f (x) - L | < ε$ .

Whew! What this is really saying is "if you tell me how close to L you want f(x) to get, then I can tell you how close to a x needs to be." This is a rigorous definition of what mathematicians mean when they say arbitrarily close—however close you want to get, I can tell you what you need to plug in to the function to get a function value that is that close.

Evaluating Limits Analytically

Basic Limits

The limit of a constant is that constant. If b is a real number, then the limit as x approaches any value a is $lim_{x \to a} b = b$ .

If a function is defined for x = a, then (usually) you can simply plug into the function to find the limit. $lim_{x \to a} x = a$ , $lim_{x \to a} x^{n} = a^{n}$ , $lim_{x \to a} \sqrt{x} = \sqrt{a}$ , etc. The main exceptions are piece-wise defined functions. Also, you can't find the limit of a radical equation at the endpoint (there aren't two sides!).

Properties

For these definitions, assume that $lim_{x \to a} f (x) = L$ , $lim_{x \to a} g (x) = K$ , and b is a real number.

Constants can be factored out: $lim_{x \to a} [b \cdot f (x)] = b \cdot lim_{x \to a} [f (x)] = b \cdot L$

Operations can be distributed: $lim_{x \to a} [f (x) \pm g (x)] = lim_{x \to a} f (x) \pm lim_{x \to a} g (x) = L \pm K$ , $lim_{x \to a} [f (x) \cdot g (x)] = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x) = L \cdot K$ , $lim_{x \to a} \frac{f (x)}{g (x)} = \frac{lim_{x \to a} f (x)}{lim_{x \to a} g (x)} = \frac{L}{K}$ (provided that K ≠ 0), and $lim_{x \to a} {[f (x)]}^{n} = L^{n}$ .

Strategies

Remember that substitution (typically) works, as long as the location of the limit (a, in the previous examples) is in the domain. Thus, the only functions whose limits will cause you any grief are those with domain problems. To date, you know of two potential domain problems: dividing by zero, and taking an even-indexed root of a negative number. Of these, the one that comes up most often is the dividing by zero issue. Let's look at how you can take care of those with some examples.

Here's a function we looked at earlier: $y = \frac{x - 2}{x^{2} - 4}$ . Now let's find $lim_{x \to 2} \frac{x - 2}{x^{2} - 4}$ .

We can't just let x = 2; we've got a domain problem. We can factor: $lim_{x \to 2} \frac{x - 2}{x^{2} - 4} = lim_{x \to 2} \frac{x - 2}{(x - 2) (x + 2)}$ . Now, remember: when we take a limit, we're letting x get really close (but at) the value where we're taking the limit. So, in this case, we want to know what happens when x is really close to (but not quite at) 2. Since we're not letting x be exactly 2, what do the factors $\frac{x - 2}{x - 2}$ do to the value of the function?

Nothing! As long as x isn't exactly 2, these factors cancel out. As far as the limit is concerned, these factors do not contribute to the function value. Cancel them out!

$lim_{x \to 2} \frac{x - 2}{x^{2} - 4} = lim_{x \to 2} \frac{1}{x + 2}$ . Now, there is no domain issue with x = 2, so we can simply plug in!

$lim_{x \to 2} \frac{x - 2}{x^{2} - 4} = lim_{x \to 2} \frac{1}{x + 2} = \frac{1}{2 + 2} = \frac{1}{4}$ .

Let's try this one (let a be a real number): $lim_{x \to 0} \frac{{(a - x)}^{2} - a^{2}}{x}$ .

Once again, we've got a domain problem at the desired value of x—we can't simply let x = 0. Let's work out that numerator: $lim_{x \to 0} \frac{{(a - x)}^{2} - a^{2}}{x} = lim_{x \to 0} \frac{a^{2} - 2 a x + x^{2} - a^{2}}{x}$ . Hey, look at that! Stuff cancels! $lim_{x \to 0} \frac{{(a - x)}^{2} - a^{2}}{x} = lim_{x \to 0} \frac{- 2 a x + x^{2}}{x}$ . Now, we can factor an x out of the numerator, which can then be canceled out (just like the last example). $lim_{x \to 0} \frac{{(a - x)}^{2} - a^{2}}{x} = lim_{x \to 0} \frac{a^{2} - 2 a x + x^{2} - a^{2}}{x} = lim_{x \to 0} \frac{x (- 2 a + x)}{x} = lim_{x \to 0} (- 2 a + x) = - 2 a$ .

Continuity and One-Sided Limits

The function f(x) is continuous at x = c provided $lim_{x \to c} f (x) = f (c)$ . So, if the limit exists (left- and right-hand limits agree), and the limit is the same as the function value, then the function is continuous at that point.

The function f(x) is continuous on the open interval (a, b) provided it is continuous at every point within the interval (a, b).

The function f(x) is continuous on the closed interval [a, b] provided it is continuous on the open interval (a, b), $lim_{x \to a^{+}} f (x) = f (a)$ , and $lim_{x \to b^{-}} f (x) = f (b)$ .

It turns out that continuity is terribly important in Calculus. Happily, most of the functions that we know and work with are (mostly) continuous.

We should be careful—we often talk about functions being continuous, when what we really mean is that the function is continuous on the set of real numbers. These two things don't mean the same thing! We, however, will not be looking into functions that are just continuous on their domains…

Infinite Limits

Sometimes, as we approach a particular value of x, the function value doesn't approach a finite value—instead, the function shoots off towards infinity. For example, the function $y = \frac{1}{x - 1}$ near x = 1.

An Image

Earlier, we noted that on the left of 1, the function heads down towards negative infinity, and that it approaches positive infinity on the right. Our definition of limits is for approaching real numbers—infinity isn't a real number. Technically, the left- and right-hand limits don't exist (the function does not approach a certain value). Nonetheless, we will write $lim_{x \to 1^{-}} f (x) = - \infty$ and $lim_{x \to 1^{+}} f (x) = \infty$ .

We can also take limits as x approaches infinity—but we'll save that for later.

Page last validated 2010-08-15