Limit and continuity

1. 1.5 and 1.6
Limits and Continuity
A. What algebra you should already know
B. What “limit” means
C. How to find a limit by direct substitution
D. How to find a limit by simplifying
E. What a “one-sided limit” is
F. What “infinite limits” are
G. What if it has 2 variables
H. What “continuous” means
I. How to determine continuities/discontinuities

2. A. What algebra you should already know for
8.1 material
• Need a reminder of any of these?
• How to find f(2) if I give you a function f(x)
• How to find f(2) if I give you a graph of f(x)
• How to graph a piecewise function
• How to simplify a rational expression
• What an asymptote is

3. B. What “limit” means
How would this be read?
( )xf
x 3
lim
→
“Approaches” means “gets closer and closer to.”

4. Let’s see if I can explain this in words.
( )xf
x 3
lim
→
This is not the same as just finding f(3).
As you can see, there is no point there.
To find the limit of f(x) as x approaches 3,
put your finger over the picture at x = 3.
With that open circle hidden, ask yourself,
“what does it look like f(3) might be if you
had to guess at it?” You would let your
eyes follow the line right on up there
really close to the hidden part. What y
value is it getting closer and closer to?
If you had the equation to work with, you
could plug in values for x like 2.5, 2.8,
2.9, 2.99, and see what the y-values are
“approaching.”

5. ( ).limFind
2
xf
x −→
Put your finger over the graph at
x = -2, and pretend like you don’t
know what it is. Look at BOTH sides
of your finger. The graph on either
side of your finger need to be
pointing at the same y value. In this
case, the two sides of the finger
seem to agree.

6. Compare: f(x) and g(x)
______)(lim
1)2(
2
=
−=−
−→
xf
f
x
___________)(lim
2)2(
2
=
=−
−→
xg
g
x

7. What if the two sides don’t agree?
( )xf
x 3
lim
→ After you give it the finger:
ANSWER:_________________________________.

8. C. How to find a limit by direct substitution
• Now that you have the idea of what the limit
is graphically, we need to learn the algebraic
ways of finding limits so that we don’t have
to graph every single function to find out
what is going on.
• When there is no funny business [holes,
jumps, asymptotes are funny business], we
can just PLUG IT IN.

9. .)(if)(limFind 2/1
4x
−
→
= xxfxf
Make sure you can do all the suggested problems. You might need to be
reminded about more algebra stuff [order of operations, negative exponents,
rational exponents, radicals, etc.]

10. D. How to find a limit by simplifying
( ) .
1
1
)(iflimFind
4
1 −
−
=
→ x
x
xfxf
x
There is a problem with “direct substitution” here!

11. ( ) .
1
1
)(iflimFind
4
1 −
−
=
→ x
x
xfxf
x

12. .
5
102
limFind:oneisYou try th
2
5 −
−
→ x
xx
x

13. E. What a “one-sided limit” is
• NOTATION:
means)"(lim"
3
xf
x −
→
means)"(lim"
3
xf
x +
→

14. )(lim
)(lim
)(lim
3
3
3
xf
BUT
xf
xf
x
x
x
→
→
→
=
=
+
−
A limit only exists if the limit from the left and the limit from the
right are equal to each other.

15. F. What “infinite limits” are
• Remember what an asymptotes is?
)(lim
)(lim
)(lim
2
2
2
xf
BUT
xf
xf
x
x
x
→
→
→
=
=
+
−

16. You try this one:
=
=
=
−→
−→
−→
+
−
)(lim
,
)(lim
)(lim
3
3
3
xf
somatchThey
xf
xf
x
x
x

17. G. What if it has 2 variables
• Same process: Hopefully you can solve it by
simply plugging it in, but you may have to
simplify a rational function.
• The difference with these is that the answer
might have a variable in it.

18. ( ).limFind 22
0
hxhx
h
++
→

19. ( ).153limFind:You try 2
0
++
→
xhx
h

20. H. What “continuous” means
• If I say, “This function is continuous at x =
2,” this is what I’m saying:
“At x = 2, this function does not have a
hole, a jump, or a vertical asymptote. It
could have holes, gaps, jumps, or
asymptote elsewhere, but at x = 2, the
function is hole-free, gap-free jump- free, and
vertical-asymptote-free.”

21. If I said, “This function is continuous,”
• “This function is continuous everywhere,
meaning there are no holes, no gaps, no
jumps, no vertical asymptotes.”
• Now that you know what a continuous graph
looks like, let’s learn how to recognize
continuity/discontinuity algebraically so we
don’t have the draw the graph for everything.

22. I. How to determine continuities/discontinuities
• All polynomial functions are “continuous”
(continuous everywhere). Examples:

23. But if the function looks like a fraction?
• Rational functions are continuous everywhere
EXCEPT where the denominator is zero.
( )
1.-at xitydiscontinuahasitEXCEPT
everywherecontinuousis
1
1
)( 2
=
+
=
x
xf

24. ( )( )
"
132
4
)(
ofitiesdiscontinuanystate
,continuousnotIf":You try
2
+−
+
=
xxx
x
xf

25. If it is piecewise?
• For it to be continuous, the graph of it has to
cover all x values, and not have any holes,
gaps, or jumps. I’ll just do these by sketching
the graph.

26. 


>−
<
=
0if,
0if,
)(
xx
xx
xf



≥−
<
=
0if,
0if,
)(
xx
xx
xg

27. Jumps are discontinuities too:
This one has a
discontinuity at
x = 3.

28. Vertical asymptotes are discontinuities.
This one has a
discontinuity at
x = 2.