Approaching ...
Sometimes we can't work something out directly ... but we can see what it should be as we get closer and closer!
Example:
(x2 − 1)/(x − 1)
Let's work it out for x=1:
(12 − 1)/(1 − 1) = (1 − 1)/(1 − 1) = 0/0
Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.
So instead of trying to work it out for x=1 let's try approaching it closer and closer:
Example Continued:
| x | (x2 − 1)(x − 1) | |
| 0.5 | 1.50000 | |
| 0.9 | 1.90000 | |
| 0.99 | 1.99000 | |
| 0.999 | 1.99900 | |
| 0.9999 | 1.99990 | |
| 0.99999 | 1.99999 | |
| ... | ... |
Now we see that as x gets close to 1, then(x2−1)(x−1) gets close to 2
We are now faced with an interesting situation:
- When x=1 we don't know the answer (it is indeterminate)
- But we can see that it is going to be 2
We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit"
The limit of (x2−1)/(x−1) as x approaches 1 is 2
And it is written in symbols as:
So it is a special way of saying, "ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2
| As a graph it looks like this:
So, in truth, we cannot say what the value at x=1 is. But we can say that as we approach 1, the limit is 2. |
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Cauchy Definition of Limit
Let f(x) be a function that is defined on an open interval X containing x=a. (The value f(a) need not be defined.)
The number L is called the limit of function f(x) as x→a if and only if, for every ε>0 there exists δ>0 such that
|f(x)−L|<ε,whenever
0<|x−a|<δ.This definition is known as ε−δ− or Cauchy definition for limit.
A GEOMETRIC EXAMPLE:
Let's look at a polygon inscribed in a circle... If we increase the number of sidesof the polygon, what can you say about the polygon with respect to the circle?
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As the number of sides of the polygon increase, the polygon is getting closer and closer to becoming the circle!
If we refer to the polygon as an n-gon, where n is the number of sides, we can make some equivalent mathematical statements. (Each statement will get a bit more technical.)
- As n gets larger, the n-gon gets closer to being the circle.
- As n approaches infinity, the n-gon approaches the circle.
- The limit of the n-gon, as n goes to infinity, is the circle!
The n-gon never really gets to be the circle, but it will get darn close! So close, in fact, that, for all practical purposes, it may as well be the circle. That's what limits are all about!
NUMERICAL EXAMPLES:
EXAMPLE 1:
Let's look at the sequence whose nth term is given by n/(n+1). Recall, that we letn=1 to get the first term of the sequence, we let n=2 to get the second term of the sequence and so on.
What will this sequence look like?
1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,... 99999/100000,...
What's happening to the terms of this sequence? Can you think of a number that these terms are getting closer and closer to? Yep! The terms are getting closer to 1! But, will they ever get to 1? Nope! So, we can say that these terms are approaching 1. Sounds like a limit! The limit is 1.
As n gets bigger and bigger, n/(n+1) gets closer and closer to 1...
EXAMPLE 2:
Now, let's look at the sequence whose nth term is given by 1/n. What will this sequence look like?
1/1, 1/2, 1/3, 1/4, 1/5,... 1/10,... 1/1000,... 1/1000000000,...
As n gets bigger, what are these terms approaching? That's right! They are approaching 0. How can we write this in Calculus language?
GRAPHICAL EXAMPLES:
On the previous page, we saw what happened to the sequence whose nth term is given by 1/n as n approaches infinity... The terms 1/n approached 0.
Now, let's look at the graph of f(x)=1/x and see what happens!
The x-axis is a horizontal asymptote... Let's look at the blue arrow first. As x gets really, really big, the graph gets closer and closer to the x-axis which has a height of 0. So, as x approaches infinity, f(x) is approaching 0. This is called alimit at infinity.
Now let's look at the green arrow... What is happening to the graph as x gets really, really small? Yep, the graph is again getting closer and closer to the x-axis (which is 0.) It's just coming in from below this time.
But what happens as x approaches 0?
Since different things happen, we need to look at two separate cases: what happens as x approaches 0 from the left and at what happens as x approaches 0 from the right:
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Since the limit from the left does not equal the limit from the right...
We need to understand how limits work, since the first part of Differential Calculus (calculus having to do with rates at which quantities change) uses them. I like to think of a limit as what the y part of a graph or function approaches as x gets closer and closer to a number, either from the left hand side which means that x part is increasing), or from the right hand side (which means the x part is decreasing).
We can write a limit where x gets closer and closer to 0 as limx→0f(x)=L. To describe this, we say the “limit of f(x) as x approaches 0 is L”. Now the beauty of limits is that x can get closer and closer to a number, but not actually ever get there (thin, or it could get there, and it would still be a limit!
The reason we have limits in Differential Calculus is because sometimes we need to know what happens to a function when the x gets closer and closer to a number (but doesn’t actually get there); we will use this concept in getting the approximation of a slope (“rate”) of a curve at that point. Sometimes, the x value does get there (like when we’re taking the slope of a straight line), but sometimes it doesn’t (like when we’re taking the slope of a curved function).
As an example, when you first learn how to handle limits, it might be the case that the x value is getting closer and closer to a number that makes the denominator of the y value 0; this can’t happen, or the fraction will “blow up”.