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Mathematical Expression Editor

We introduce limits.

The basic idea

Consider the function 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 is 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 allow x values on the left of 2 to get closer and closer to 2, we see that .
However, if we allow the values of x on the right of 2 to get closer and closer to 2 we
see

So just to the right of 2, . We cannot find a single number that approaches as
approaches 2, 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 rounding to three decimal places: We may rush
and say that, based on the table above, But, recall the definition of the limit: is the
limit if the value of is as close as one wishes to for all sufficiently close, but not
equal to, .

From this table we can see that is as close as one wishes to for some values that
are sufficiently close to . But this does does not satisfy the definition of the limit, at
least, not yet.

But, wait! Fill in another table. What do these two tables tell us about

The limit does not exist.

The limit does not exist. The first table shows that we can always find a value
of as close as we want to such that . However, the limit is not equal to ,
since the second table shows that we can also find a value of as close as we
want to such that . It turns out that for any number , , we can find a value
of as close as we want to such that . Check the graph of the function .

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, its graph looks much “nicer” near than does
the previous graph near . More can be said about the function and its behavior near
.

Intuitively,

for the function , is the limit from the right as approaches ,

written if the value of is as close as one wishes to for all sufficiently close to
.

Similarly,

for the function , is the limit from the left as approaches ,

written if the value of is as close as one wishes to for all sufficiently close 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.