Calculus! #16: Early Transcendentals 2.2A

Okay, so now that you have the basic idea of what a limit is, we’re going to develop your intuition a little.

Section 2.2: The Limit of a Function

This whole chapter is to show you two things: 1) How a limit works on a function in general, 2) how to deal with limits in practice.

So, let’s start off with the book’s definition of a limit

lim_{x \to a} f(x) = L

Verbally, this is “the limit of f(x) as x approaches a equals L.” What it means is that for a function f(x), the closer we take x to a certain value, a, the closer the limit will get to L.

In other words, for some function f(x) = x, as we take x to one value, f(x) approaches some other value. That’s the whole idea of a limit: As the input of a function approaches some value, the output of that function approaches some other value.

Let’s go through the book’s examples and develop some intuition.

Example 1:

Guess the value of \lim_{x \to 1}\dfrac{x - 1}{x^2 - 1}.

So, take this mathematical expression and figure out what it approaches as x edges closer and closer to 1. You’ll note that you can’t just put in the 1 and get a value out. That’s what makes this a limit problem. If you wanted the limit as x approaches 1 of the function “f(x) = x+1,” it’d be easy. Just substitute 1 in for x and you get 2.

In the case of this example it doesn’t work, because the function approaches 0/0, which is undefined. So, as we did in previous problems we take x closer and closer to 1 without actually hitting 1 and see what the result seems to approach. In the book, they go as close as .9999 for x, and get a value of 0.500025 for f(x). This suggests that the value is 1/2.

In addition, they do values from the other direction, getting as close as 1.0001. At that x value, the f(x) value is .499975. So, it seems very likely that the limit’s value is 0.5. As we edge closer and closer to x=1, f(x) gets closer and closer to 1/2.

It’s important to note here that 0.5 is the limit, EVEN THOUGH there is not a defined value at x=1. For the graph, we’d put a little hole at x=1 because it’s undefined. However, the limit isn’t a measure of what happens when x equals something. It’s a measure of what happens as x approaches something.

Are you starting to get an intuitive feel for these? Let’s do another example.

Example 2:

Estimate the value of \lim_{t \to 0} \dfrac{\sqrt{t^2 + 9} -3}{t^2}.

Once again, note that you can’t just pop the value t approaches and get a real answer. You’ll get 0/0.

See if you can solve this one yourself, then I’ll give you the answer.

Did you solve it?

LIAR!

Okay, how about now?

Okay, good.

LIAR!

As you get closer and closer, you approach 0.1666666…, and the closer you get, the more sixes you can pop on. So, it seems very likely that you’re approaching 0.1 followed by infinity sixes, which is also known as 1/6.

Here, the book makes a side point about the pitfalls of computer-based calculations. For the system they’re using, at about t=0.00005 they start getting a value of 0 for f(x). This is a good example of why it’s always good to know the math “under the hood.” No computer is accurate to infinite decimal places. At a certain point, it’s just rounding. In this case, when the decimal (after you square t) gets to be on the order of billionths, it says “fuck it,” turns it into a 0, and gives you the wrong value.

So far, the limits we’ve done have been pretty intuitive. Let’s  look at one that might surprise you:

Example 3

Guess the value of \lim_{x \to 0} \dfrac{sin x}{x}

What’s your intuitive guess? Maybe you think it’s 0, since you know the numerator goes to 0 as x goes to 0. But, that doesn’t work because the denominator does too. Maybe you think limits don’t make sense for periodic functions. Also wrong – remember, we’re approaching a particular point. It doesn’t matter how the function behaves elsewhere.

So, let’s go back to calculating the actual values. When we do, we find that the closer x gets to 0, the closer the function gets to 1.

This may seem like a small thing, but it’s a big deal in physical calculations. It means that, as the physicists say “for small values of sin(x), sin(x) = x.” That’s a big deal. Imagine you’ve got an ugly equation with the sine of some big pile of variables. Now, imagine you can remove the sin() part. A common physics example is pendular motion. Part of the calculation for how a pendulum moves involves the maximum angle of swing it achieves. If you can just use the value of the angle, rather than the sine of the angle, it massively simplifies things.

Of course, this is a bit of a rule of thumb, so it’s arbitrary as to exactly what “small” means. The version I was taught is that you’re good down to around 15 degrees (π/12 radians).

So, you can already see how limits are helping us out. Hopefully, you can also see how limits can give unintuitive results.

With that in mind, let’s test your intuition again!

Example 4

What is the value of \lim_{x \to 0} sin{\dfrac{\pi}{x}} ?

In this case, as x gets smaller and smaller, you’re taking the sine of a larger and larger number. As you know, sine is a periodic function, meaning that it wobbles up and down as you walk down x. So, in this case, you have a problem. As x goes to 0, the function operates on bigger and bigger values. But, as those values get bigger and bigger, the operation (sine) stays between -1 and 1, wobbling back and forth forever. So, there is no particular value you approaches as π/x gets bigger.

Therefore, we say that the limit does not exist.

One more test of your intuition!

Example 5

Find \lim_{x \to 0} (x^3 + \dfrac{cos{5x}}{10,000}).

Now, say you make a list of what happens as x goes to 0. You’ll note that it seems to be getting smaller and smaller, approaching 0. So, you might guess that the function approaches 0. BUT YOU’VE BEEN PLAYED FOR THE FOOL, MY FRIEND.

Look at the function again. We know for sure that the left part, x^3 simply goes to 0 as x goes to zero. What about the right part? Well, as x goes to zero, cosine goes to 1. So the right part goes to 1/10,000. So, as x gets closer and closer to 0, our function should actually approach 1/10,000. That is to say, it gets very small indeed, but it does not reach 0. You can confirm this by graphing it and seeing if the function ever touches zero. It doesn’t.

The lesson here is this: You can’t just look at a list of numbers and assume they’ll lead you to the limit. That list of numbers is just an intuitive way to look at things. Getting 0 instead of 1/10,000 is pretty good. In fact, it’s only off by 1/10,000. But, in the right context, that might matter quite a bit. What if the equation predicts what percent of people will die of ultra-plague when I release it later this year? At 1/10,000, you’re looking at 600,000 dead people – all of them dead because you didn’t understand the concept of a limit.

All these examples may seem a bit different, but they’re getting at the same idea – limits are what f(x) approaches as x approaches something.

Unfortunately, it’s not always quite that simple…

Example 6

The Heaviside Function is given by H(t). H(t) is 0 when t is less than zero, and 1 when H(t) is greater than or equal to zero. That bastard got a whole function named after him that’s simple enough to be in a pre-calc text.

Take a look at the link there, which shows a graph of the function. What do you think the limit is as you get closer to 0?

You’ll immediately see a problem. If you approach from one side, the limit is 0. If you approach from the other side, the limit is 1. So, it’s not clear that there’s a single limit. But, you’re not as lost as in example 4, where there was no limit at all. Here, you can at least say there seem to be 2 limits.

And, you’d be right to say that. In fact, many equations have more than 1 limit. That’ll set us up for the next blog, on One-sided Limits.

 

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8 Responses to Calculus! #16: Early Transcendentals 2.2A

  1. Niznar says:

    Looks like a small typo in Example 4, pi/x isn’t inside of a sine function, although in both cases the limit is undefined.

    Love this series, I’m going to recommend it to the students I tutor

  2. Stobes says:

    Think there’s a typo in example 4. From the explanation it seems sin(pi/x) was intended as the function.

  3. david says:

    “In other words, for some function f(x) = x,” i think you meant f(x)=y.

    also in the cosine function you wrote “So, as x gets closer and closer to 1,” but it was lim x->0

  4. Dan says:

    Might be handy to have a couple sample problems at the end of some of the posts. Dunno if you were already planning on doing that in the future, but these are a good refresher and a few problems to solve would help.

    • ZachWeiner says:

      I’m not planning on that at this time. These posts are meant to be a helpful guide to a particular book, which is already full of examples :). Also, although I’m doing this publicly, this is mostly for my personal self cultivation. I just figured other people could benefit, and having to explain things would improve my learning.

  5. I tried to follow this but it just gave me nightmares like this: http://madartlab.com/2011/08/15/ai-did-your-math-class-look-like-this :-D

  6. Eric says:

    Typo!

    In defining the Heaviside function, you say
    “1 when H(t) is greater than or equal to zero” but you mean
    “1 when t is greater than or equal to zero”

    Otherwise: Recursive nightmare!

    Keep posting, this is good stuff.

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