2.7 Limits involving infinityThurs Oct 1

Do Now

Find

Undefined limits

• Let us examine

• Recall that it is undefined at x = 0, so we say that it does not exist.

• We know there’s an asymptote

• If asymptote =

•From the left side, f(x) decreases without bound. We can say that

•From the right side, f(x) increases without bound. We can say that

One-sided Limits that agree• If both one-sided limits agree, we can

say that the whole limit exists. This is now true for infinity.

• Ex:

Ex 2

You try

Find the limits

Determining the sign of infinity• To determine whether it is positive or

negative infinity, plug a # very close. We write a + or - above/below each factor.

• Example:

Limits Approaching Infinity

• We are also interested in examining the behavior of functions as x increases or decreases without bound ( )

Thm- For any rational number t > 0,

as long as there isn’t a negative in a radical

Thm- For any polynomial of degree n > 0, and

then

Limits to infinity with a rational function

• When the limit is approaching infinity and the function is a fraction of two polynomials, we have to look at the power of the numerator and denominator (The highest exponent in each).

• We divide every term by the variable with the highest power

• There are 3 cases

Limits to infinity with a rational function pt2

• Case 1– The power of the numerator is bigger

In this case, the limit = +/- infinity

Example: Evaluate

Limits to infinity with a rational function pt3

• Case 2:– The power of the denominator is bigger

In this case, the limit = 0

Example: Evaluate

Limits to infinity with a rational function pt4

• Case 3:– The powers are the same

In this case, we must look at the coefficients of the first terms. The limit = the fraction of coefficients

Example: Evaluate

You TryEvaluate each limit1)

2)

3)

Closure

• What happens when we have a rational function and the limit approaches infinity?

• Describe the 3 cases

• HW p. 105 #7-15, 23-27 odds

2.7 Limits to infinityFri Oct 2

• Do Now

• Find each limit

HW Review:p105 #7-15 23-27• 7) 1 • 9) 0• 11) 7/4• 13) - infinity• 15) + infinity• 23) 0• 25) 2• 27) 1/16

More book ex if necessary

Limits approaching infinity

• What about rational functions that aren’t only polynomials?

• Trig functions: they typically do not exist.

• Ex: sin x

Exponential functions

• Notice exponential functions increase much faster than algebraic polynomials. Consider their ‘exponents’ as greater than any polynomial’s exponent.

• Ex:

e^x increases faster, so the limit = 0

Logarithmic Functions

• The inverse of the previous slide is also true.

• Logarithmic functions increase very slowly as x approaches infinity

• Therefore, consider their “exponents” as less than any algebraic polynomial.

• Ex:

Classwork

• (green book) Worksheet p.122 #31-40

Closure

• Journal Entry: What did we learn about functions with numerators and denominators that are not polynomials?

• How do we determine those limits as x approaches infinity?

• HW: Finish worksheet p. 122 #31-40

2.7 Limits Involving InfinityMon Oct 5

Do NowEvaluate each limit

1)

2)

3)

HW Review: worksheet p.122 #31-40

31) infinity 36) 0

32) 0 37) infinity

33) DNE 38) neg inf

34) DNE 39) 0

35) 0 40) infinity

PracticeWorksheet 5-28

ClosureHand in: Find the limits, if they exist

1)

2)

HW: Finish worksheet p.131-132 #7-28

2.7 ReviewWed Oct 2

HW Review worksheet p.131-132 #7-28• 7) neg infinity 18)

• 8) infinity 19)

• 9) infinity 20)

• 10) infinity 21)

• 11) 3/2 22)

• 12) 5/2 23)

• 13) 0 24)

• 14) 0 25) neg infinity

• 15) 0 26) infinity

• 16) 5/3 27) -1/7

• 17) 28) 4/7