1. How do you confirm when you have a vertical or a

horizontal asymptote

Thought of the Day

Vertical – suspect them at x-values which cause the denominator of f(x) to be zero.

Confirm that the limit as x a is infinite….

Horizontal – suspect rational functions

Confirm that as x , y a

What is the definition of LIMIT?

OK…this is like the basis of ALL of Calculus.

It was finally “perfected” by Cauchy in 1821.

Ready?

Given any if there is a corresponding such that

implies then we say that

lim

0 0

ax0 Lxf )(

Lxf )(ax

(This is the bare bones important part that you need to memorize…check your text for the detailed version.)

How many different methods are there for

evaluating limits?

Can you name several?

1. Inspection

2. Observe graph

3. Create a table of values

4. Re-write algebraically

5. Use L’Hopitals Rule (only if the form is indeterminate)

6. Squeeze theorem (rarely used!!)

How many indeterminate forms can you name?

00

00

0

1

0

Did you know all 7?

1.

2.

3.

4.

5.

6.

7.

Math Wars!!!

limxxsin = ?

0x

1

lim0x

?cos1

xx

0Zero! Zip…

What are the three main types of discontinuities?

1. Hole – at x=3 in the example

2. Step – usually the function’s description is split up :

3. Vertical asymptote – at x=1 in the example

)3)(1()3(

xx

x

2xx2

for x<0

for x>0f(x)={

Under what conditions does the derivative NOT exist at x=a

If there is a discontinuity at x=a or if there is a sharp corner at x=a, then the

derivative is undefined at x=a

What is the definition of continuity at a point?

lim)(af )(xfax

What is a monotone function?

A function that is either always increasing or always decreasing.

(i.e. the derivative is always positive or always negative.)