Limits

　

We have learned that a line L is tangent to the circle at a point P if L passes through P perpendicular to the radius at P , as shown in the graph below.

　

What does it mean to say a line L is tangent to some other curve C ? To define tangency for general curves, we need a dynamic interpretation of the tangency for a circle. The tangent line to a circle at a point P can be viewed as a result of the secant lines through P and near by points Q as Q moving toward P along the circle, as show in the animations below. (A line joining two points on the curve is a secant line to the curve.)

　

 

　

 

　

Suppose we want to find the equation of the tangent line to the parabola at the point . Since the tangent line passes through , we will be able to to get the equation of the tangent line as soon as we know its slope . The difficulty is that we need two points to compute the slope. However, we can compute an approximation to by choosing a nearby point , where , on the parabola and compute the slope of the secant line .

　

Here we compute the values of for several values of close to .

　

　

It appears from the table above that the closer is to , the closer is to , the closer is to . It suggests that the slope of the tangent line should be . (We shall discuss more about tangent lines in a later secession.)

　

 

　

 

Definition of a Limit

　

Given a function defined for all near , except possibly at . We say that

　

" the limit of f( x ), as x approaches a , equals L "

　

if we can make the values of f( x ) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a , but not equal to a . In this case, we write

　

　

We see from the table that the values of can get quite close to , can we make the values of as close to as we like by taking to be sufficiently close to ?

　

If we want the distance between and to be less than , how close to do we have to take ? And what happens if we want the distance between and to be less than ?

　

In other words, how close to do we have to take so that

　

( or ) ?

　

Notice that is not defined when and when . Hence, we have

　

　

which is the distance from to . For the first case, we can simply take so that 0 < , that is, the distance from to is less than . For the second case, we can take so that 0 < .

 

Similarly, if we want the distance between and to be less than a given number , we can take so that 0 < . Therefore,

　

　

From the argument above, we see that does not depends whether is defined at nor how is defined at . Consequently, given any number , if function is defined by

　

if and ,

then

　

In particular, we have

　

　

　

Example 1 Investigate .

　

　

　

The graph above suggests that the values of approaches as approaches . We guess that

　

　

How close to do we have to take so that ?

Notice that if and only if . That means we can take so that . Geometrically, we can also see this by the graph below.

　

　

Since is a line of slope , we can see from the graph above that it suffices to take so that

　

　

Moreover, if we want the distance between and to be less than a given number , we can take so that 0 < . Therefore, .

　

　

Example 2 Investigate .

Let's take a look at the values of for several values of close to . Since is even, we can take only positive values of .

　

　

　

As approaches , the value of the function seem to approach 0 ... and so we guess that

　

　

What would have happened if we had taken even smaller values of ?

　

　

Here is a graph of for t close to 0 within 0.01.

　

What happen to the plot ?? Does this mean that the limit is instead of ????

　

Notice that is not defined when and when .

　

　

　

　

So the limit should be . What was wrong with previous calculations ? ----- The CAS (Maple) gave false values due to round up errors.

　

Can we make the values of as close to as we like by taking to be sufficiently close to ? How close to do we have to take so that

　

　

　

Hence, we can take so that ( or even smaller). But this seems to be hard way to confirm that

we shall do it in another way later.

　

Example 3 Investigate .

　

Suppose that , for some number (in this case, we say exists.). Since the value of is always as approaches from the right, . But the value of is always as approaches from the left, so . Here we get a contradiction. Therefore, does not exist.

　

One-sided limits

　

We say the limit of as approaches from the left is equal to and write

　

　

if we can make the values of f( x ) arbitrarily close to L by taking x to be sufficiently close to and .

　

similarly, w e say the limit of as approaches from the right is equal to and write

　

　

if we can make the values of f( x ) arbitrarily close to L by taking x to be sufficiently close to and .

　

With these notations, we have

and

　

By comparing the definition of limits with the definitions of one-sided limits, we see that

　

if and only if and

　

Example 4 Investigate .

　

　

Since is an even function, we can just look at the values of for . We see from the table above that as gets close to , gets close to , and gets very large.

In fact, given a positive number ( no matter how large is), if we take then .That means the values of can be made as large as we like by taking close enough to .

　

Infinite Limits

　

Let be a function defined on both sides of , except possibly at . Then we say the limit of is infinity as approaches , and write

　

　

if the values of can be made as large as we like by taking close enough to , but not equal to .

W e say the limit of is negative infinity as approaches , and write

　

　

if the values of can be made as large negative as we like by taking close enough to , but not equal to . Similar definitions can be given for the one-sided infinite limits

　

　

　

　

From the previous discussion, we see that . Let's take a look at the graph of .

　

It appears from the graph above that the graph of gets closer to the y -axis as approaches to .

　

 

　

Vertical Asymptotes

　

The line is called a vertical asymptote of the curve if at least one of the following statements is true :

　

　

　

(a) , .

　

　

　

(b) , .

　

　

(c) .

　

　

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