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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: 1 lim 2 = +¥ x®0 x Ex 2 You try Find the limits 1 - 2x lim+ 2 x ®1 x - 1 4- x lim+ 2 x ® 2 ( x - 2) 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 ( x ® ±¥) 1 lim = 0 x ®¥ x 1 lim = 0 x ® -¥ x Thm- For any rational number t > 0, 1 lim t = 0 x ® ±¥ x as long as there isn’t a negative in a radical Thm- For any polynomial of degree n > 0, and p(x) = an x n + an -1x n -1 + ...+ a0 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, 23-27 odds 2.7 Limits to infinity Fri Oct 2 • Do Now • Find each limit - 3x + 2 x - 4 lim x ®¥ 5 x 3 + x 2 + 1 3 lim x ®¥ -x 4+ x 2 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: x lim x ®¥ ln x =¥ 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 #31-40 31) infinity 36) 0 32) 0 37) infinity 33) DNE 38) neg inf 34) DNE 39) 0 35) 0 40) infinity Practice Worksheet 5-28 Closure Hand in: Find the limits, if they exist 1) 2) HW: Finish worksheet p.131-132 #7-28 2.7 Review Wed Oct 2 • • • • • • • • • • • HW Review worksheet p.13113218)#7-28 7) neg infinity 3/2 3 8) infinity 9) infinity 10) infinity 11) 3/2 12) 5/2 13) 0 14) 0 15) 0 16) 5/3 17) -3 5 /8 19) - 5 20) 5 6 1 21) 6 = 6 6 22) - 16 = - 6 23) 3 24) 3 25) neg infinity 26) infinity 27) -1/7 28) 4/7