2.7 Limits involving infinity Thurs 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 Try Evaluate each limit 1) 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, odds

2.7 Limits to infinity Fri Oct 2 Do Now Find each limit

HW Review:p105 # ) 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 Infinity Mon Oct 5 Do Now Evaluate each limit 1) 2) 3)

HW Review: worksheet p.122 # ) infinity36) 0 32) 037) infinity 33) DNE38) neg inf 34) DNE39) 0 35) 040) infinity

Practice Worksheet 5-28

Closure Hand in: Find the limits, if they exist 1) 2) HW: Finish worksheet p #7-28

2.7 Review Wed Oct 2

HW Review worksheet p #7-28 7) neg infinity18) 8) infinity19) 9) infinity20) 10) infinity21) 11) 3/222) 12) 5/223) 13) 024) 14) 025) neg infinity 15) 026) infinity 16) 5/327) -1/7 17) 28) 4/7