1.5 and 1.6 Limits and Continuity

A. What algebra you should already knowB. What “limit” meansC. How to find a limit by direct substitutionD. How to find a limit by simplifyingE. What a “one-sided limit” isF. What “infinite limits” areG. What if it has 2 variablesH. What “continuous” meansI. How to determine continuities/discontinuities

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

B. What “limit” means

How would this be read?

( )xfx 3lim

→

“Approaches” means “gets closer and closer to.”

Let’s see if I can explain this in words.

( )xfx 3lim

→

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.”

( ).lim Find2xf

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.

Compare: f(x) and g(x)

______)(lim

1)2(

2=

−=−

−→xf

f

x

___________ ) ( lim

2 ) 2 (

2=

=−

−→xg

g

x

What if the two sides don’t agree?( )xf

x 3lim

→ After you give it the finger:

ANSWER:_________________________________.

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.

.)( if )(lim Find 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.]

D. How to find a limit by simplifying

( ) .1

1)( if lim Find

4

1 −−=

→ x

xxfxf

x

There is a problem with “direct substitution” here!

( ) .1

1)( if lim Find

4

1 −−=

→ x

xxfxf

x

.5

102lim Find :one isYou try th

2

5 −−

→ x

xxx

E. What a “one-sided limit” is

• NOTATION:

means )"(lim"3

xfx −→

means )"(lim"3

xfx +→

)(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.

F. What “infinite limits” are

• Remember what an asymptotes is?

)(lim

)(lim

)(lim

2

2

2

xf

BUT

xf

xf

x

x

x

→

→

→

=

=

+

−

You try this one:

=

=

=

−→

−→

−→

+

−

)(lim

,

)(lim

)(lim

3

3

3

xf

somatchThey

xf

xf

x

x

x

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.

( ).lim Find 22

0hxhx

h++

→

( ).153lim Find :You try 2

0++

→xhx

h

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.”

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.

I. How to determine continuities/discontinuities

• All polynomial functions are “continuous” (continuous everywhere). Examples:

But if the function looks like a fraction?

• Rational functions are continuous everywhere EXCEPT where the denominator is zero.

( )1.- at x ity discontinu a hasit EXCEPT

everywhere continuous is 1

1)( 2

=+

=x

xf

( ) ( ) "132

4)(

of itiesdiscontinuany state

,continuousnot If" :You try

2 +−+=xxx

xxf

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.

>−<

=0 if ,

0 if ,)(

xx

xxxf

≥−<

=0 if ,

0 if ,)(

xx

xxxg

Jumps are discontinuities too:

This one has a discontinuity at x = 3.

Vertical asymptotes are discontinuities.

This one has a discontinuity at x = 2.