1. 16.1 Limit of a Function of a Real Variable

2. (Don’t write – just explore)
Let’s look at a graph and look at its table of values.
Desmos…. In box 1, type f (x) = 0.5x + 3
In box 2, go to & choose table
In y spot, delete y and type f (x)
1 is already there, let’s do some more (type in x column)
1.5
1.75
1.85
1.95
1.999
2.001
2.05
2.15
2.25
Now, look at the graph & the table of values
As the x-values get closer to 2, what do the y-values get close to?
So… as x approaches 2, y approaches 4 4
*this idea has a name…. but something to watch first…

4. Why does she say “the limit does not exist?”
Back to Desmos….
Delete equation in box 1 and type
Look at this graph & the table of values
Answer the same question as before…
As the x-values get closer to 2, what do the y-values get close to?
Discuss with a partner
It depends on if you are approaching 2 from the left or the right
From the left  y-values get very large
From the right  y-values get very small
So, “the limit does not exist.”
Let’s ask another question … As the x-values get SUPER BIG or
SUPER SMALL, what do the y-values get close to? 2

5. These ideas of the x-values getting close to something and the resulting y-values getting close to something is the idea behind a LIMIT.
(Now write)
Notation:
Ex 1) Using the graph (on Desmos) of each function, find the limit.
gets way big
gets way small

6. Sometimes as x approaches a specific x-value from the right (means traveling left) or from the left (means traveling right), the y-values do not match – that is ok!
(from the right)
(from the left)
If they are the same, ,
we say the limit exists at a and
(*Special note: the function may or may not have the same value as the limit at a particular a – it’s ok!)

7. Ex 2)
the limit exists!
If you are dealing with a polynomial function (like we did at the beginning of the lesson), the value of the limit IS the value of the function.
If a function f (x) is a polynomial function on an interval (a, b) and c is a real number on that interval, then

8. Ex 3) Find the limit of each function.
(plug it in) 14 2
We may also have to examine a graph to find a limit.
Ex 4)
at 0, this function is undefined,
but as it approaches 0 from the right and the left, it approaches the y-value of 1
*graph on Desmos

9. A precise formal mathematical definition of a limit is as follows:

10. Homework
#1601 Pg 857 #1, 3, 5, 7–14, 17, 19, 21, 23, 25, 26, 30, 32–36, 38, 41–44