We introduce limits.

The basic idea

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

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Use the graph of above to answer the following question: What is ?
If you are struggling to determine , you may want to review how to evaluate functions using a graph here or here.
is undefined it 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.

Use the graph of above to finish the following statement: ‘‘A good guess is that…’’
. . . .
Consider the following graph of

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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.

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

Consider . Fill in the tables below: What do the tables tell us about
it is unclear what the tables are telling us about the limit

One-sided limits

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

When you put this all together

One-sided limits help us talk about limits.

Evaluate the expressions by referencing the graph below. Write DNE if the limit does not exist.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)