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Sample of Grade 12 Notes - LIMITS

Limits


Limits are actually one of the easier concepts of Calculus to understand. They are of two varieties: directional and non-directional. A non-directional limit would take the form of:

Lim (x2 - 1)/(x - 1)
X->1

A directional limit would take the form of:

Lim (x2 - 1)/(x-1)
x->1+

It just so happens that the answer to both of these limits have the same answer; however, be cautioned that this will not always be the case. The first example is an absolute limit, the second is a limit taken as the function approaches from the right (or positive) side of the point. If it had been a - instead of a +, then it would have approached from the left (or negative) side of the point.

A simple definition for a limit is the value that a function approaches as the value of the variable gets near the given point.

In the example above, we are asking, "What value does (x2-1)/(x-1) approach as x gets near 1. All good Algebra I students would tell you that this value can not be determined because x-1=0! However, with Calculus and the use of limits, we can determine what value it will approach.

WORD OF CAUTION: The value of (x2-1)/(x-1) is still undeterminable because we can NOT divide by zero.

As to the working of this problem, there are several ways that it can be done:

  1. Make a chart of values
  2. Graph
  3. Plug it in
  4. Use L'Hôpital's rule


Make a chart of values

If we were to put various values in for x, we would come up with the following table:

X

Value

1.12.1
1.012.01
1.0012.001
1.00012.0001
1.000012.00001
1.0000012


Because of the precision that my calculator was using, we got an imprecise value for the last one, it should have been 2.000001, but we can see the trend very clearly. As x gets close to 1, the value gets closer to 2. We can deduce from this that the limit of (x2-1)/(x-1) as x approaches 1 from the right is equal to 2.

Notice, however, that the above chart is incomplete. This is due to the fact that our chart only considers the values from the RIGHT side of 1. In order to complete the non-directional limit, we would also have to find:

X

Value

.91.9
.991.99
.9991.999
.99991.9999
.999991.99999
.9999991.999999
.99999991.9999999


The values in this chart provide us with the limit from the left as being equal to two. Since the directional limit from the left and the right are both equal to two, then the non-directional limit is also equal to two.

As one can see, this is probably not the way that you want to do all of your limits. It is important that you consider both directions in a non-directional limit because sometimes they do not match. When this happens then it is said that the non-directional limit "does not exist."


Graph

The second method of determining limits (and probably the most unreliable one) is to graph the equation. I say that it is unreliable for many reasons.

  1. Our eyes can deceive us.
  2. Human error is quite common, which can and will cause an error in our graph
  3. Neatness is a HUGE factor.

However, if we are not looking for precision, then graphing is a method that we can use. Let us consider the example that we have been using. If we were to graph it, we would get:

By examining this graph, one can tell that as x gets closer to 1, the value of the expression in question gets closer to 2. This happens from both the left and the right side of x=1, so all three of the limits (left, right, and non-directional) for this expression at 1 are equal to 2.


Plug it in

Of course, this is not the most technical way of saying it, but substituting the value in for x will often work. In most cases, however, there is some additional work to be done first. Most definitely in this case since the denominator is equal to 0. Consider this, though:

(x2-1)/(x-1)=(x-1)(x+1)/(x-1)=(x+1)

It is a very simple matter to now substitute a 1 in for x. At x=1, x+1=2.

Notice that we got the same limit as we did using the other two methods. It still remains, however, to determine if that is a non-directional limit or a directional one. This can easily be accomplished by sketching a quick graph or substituting values on either side of 1 as was done in the previous two methods.




Use L'Hôpital's Rule



Warning: If you have not studied derivatives, do not read this. The use of this method requires knowledge of derivatives, and if you have not yet studied derivatives, then it may only serve to confuse you. Please read the section on derivatives first and then return. (That should take a while.)

It happens on occasion that the expression can not be simplified as was done in the last section. When that happens, there is an alternative to tables or graphing and that alternative is L'Hôpital's rule.

L'Hôpital's Rule: If the expression in a limit is of the form f(x)/g(x) and evaluates to 0/0 or / at the point in question, then the limit is equal to the limit of f'(x)/g'(x) at the same point.

Simply stated, if when we plug the value in, we get 0/0 or /, then we can take the derivative of both the numerator and the denominator and try again. If it still can not be resolved then this rule can be used again.

Do NOT immediately use L'Hôpital's rule! The best method is still to simplify and plug in. Although I can not think of an instance where it will give an incorrect result, you should still try to simplify first. Don't take the lazy way out. For the sake of argument, however, let's consider our problem.

F(x)=x2-1 G(x)=x-1

F'(x)=2x G'(x)=1

Therefore, it becomes a simple matter of:

Lim 2x
X->1

Plugging it in, we get, 2(1) = 2.

Consider this problem:

Lim sin(x)/cos(x-pi/2)
X->0


A simple way to substitute into this problem is probably not immediately evident to most people (although the substitution does exist), so many would jump straight to the rule.

F(x) = sin(x) G(x)=cos(x-pi/2)

F'(x) = cos(x) G'(x)=-sin(x-pi/2)


Which leads us to:

Cos(0)/-sin(-pi/2) = 1/-(-1) = 1.

I hope that this discussion has helped somewhat with any problems that you might be having. If additional help is needed, you can email me.
What's a Limit?

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(This lesson will explain the concept of a limit from various points of view.)

A GEOMETRIC EXAMPLE:

Let's look at a polygon inscribed in a circle... If we increase the number of sides of the polygon, what can you say about the polygon with respect to the circle?

triangle in circlesquare in circle
hexagon in circledecagon in circle

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!

Archimedes used this idea (WAY before Calculus was even invented) to find the area of a circle before they had a value for PI! (They knew PI was the circumference divided by the diameter... But, hey, they didn't have calculators back then.)

 

SOME NUMERICAL EXAMPLES:

EXAMPLE 1:

Let's look at the sequence whose nth term is given by n/(n+1). Recall, that we let n=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?

What if we stick an x in for the n? Maybe it will look familiar... Do you remember what the graph of f(x)=1/x looks like? Keep reading to see our second example shown in graphical terms!

SOME 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 a limit at infinity.

The limit of 1/x as x approaches infinity is 0

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.

The limit of 1/x as x approaches negative infinity is 0

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:

     and     

Since the limit from the left does not equal the limit from the right...

Let's look at a more complicated example...

Given this graph of f(x)...

First of all, let's look at what's happening around the dashed blue line. Recall that this is called a vertical asymptote.

           

 

So...     

Another way to think about the limit is the find the height of the graph at (or really close to) the given x. Think about a little mountain climbing ant (call him Pierre) who is crawling on the graph. What is Pierre's altitude when he's climbing towards an x? That's the limit!

Let's try some more...

Let's look at what's happening at x = -7... The limit from the right is the same as the limit from the left... But there's a hole at x = -7!

That's ok! We don't care what happens right at the point, just in the neighborhood around that point. So...

Can you find the limit of f(x) as x approaches -3?

That's right!     

How about the limit of f(x) as x approaches 0?

Right again!     

Let's look at one more type of limit. To do this we'll show you the screen of a TI-92 graphing calculator!

This is the graph of

TI-92 graph

(If you have a TI-92, the viewing window here is -1.5, 2.1, 1, -3.3, 4.7, 1, 2.)

It sure wiggles around a lot! But, we see that

Well, that's all I have to say about limits right now! I hope it helped. If not, go listen to some good Calculus music...

Take It To The Limit by The Eagles!

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