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Mathematical Expression Editor

We introduce limits.

The basic idea

Consider the function graphed below. While is undefined at , we can still plot at
other values near .

Use the graph of above to answer the following question: What is ?

is
undefinedit is impossible to say

Nevertheless, we can see that as approaches zero, approaches one. From this setting
we come to our definition of a limit.

Intuitively,

the limit of as approaches is ,

written if the value of can be made as close as one wishes to for all sufficiently
close, but not equal, to .

Use the graph of above to finish the following statement: “A good guess is that …”

....

Consider the following graph of

Use the graph to evaluate the following. Write DNE if the value does not
exist.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Limits might not exist

Limits might not exist. Let’s see how this happens.

Consider the graph of .

Explain why the limit does not exist.

The function is the function that
returns the greatest integer less than or equal to . Recall that if can be
made arbitrarily close to by making sufficiently close, but not equal to, .
So let’s examine near, but not equal to, . Now the question is: What is
?

If this limit exists, then we should be able to look sufficiently close, but
not at, , and see that is approaching some number. Let’s look at a graph:

If we look closer and closer to (on the left of ) we see that . However, if we look
closer and closer to (on the right of ) we see

So just to the right of , . We cannot find a single number that approaches as
approaches , and so the limit does not exists.

Tables can be used to help guess limits, but one must be careful.

Consider . Fill in the tables below (to three decimal places): What do the tables tell
us about

it is unclear what the tables are telling us about the limit

Neither tables nor graphs can ever tell us for certain what a limit is. However,
sometimes they can help “guess” the limit. In this case the graph of is somewhat
more helpful:

We see that oscillates “wildly” as approaches , and hence does not approach any one
number.

One-sided limits

While we have seen that does not exist, more can still be said.

Intuitively,

the limit from the right of as approaches is ,

written if the value of can be made as close as one wishes to for all sufficiently
close, but not equal to, .

Similarly,

the limit from the left of as approaches is ,

written if the value of can be made as close as one wishes to for all sufficiently
close, but not equal to, .

Compute: by using the graph below

From the graph we can see that as approaches from the left, remains at up until
the exact point that . Hence Also from the graph we can see that as approaches
from the right, remains at up to . Hence

When you put this all together

One-sided limits help us talk about limits.

A limit exists if and only if

exists

exists

In this case, is equal to the common value of the two one sided limits.

Evaluate the expressions by referencing the graph below. Write DNE if the limit
does not exist.