Math Calculus f. Business and Management - Worksheet 12. Solutions for Worksheet 12 - Limits as x approaches infinity

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1 Math 0 - Calculus f. Business and Management - Worksheet 1 Solutions for Worksheet 1 - Limits as approaches infinity Simple Limits Eercise 1: Compute the following its: 1a : + 4 1b : c : Solution to #1: For 1a: + 4 = ( ) + 4 =, since both and ( ) go to infinity. For 1b: = infinity. ( ) + 8 =, since both and ( ) go to infinity as goes to For 1c: = ( ) + 8 =, since goes to and ( ) goes to. Eercise : Compute the following its: a : 4 b : +4 4 c : e Solution to #: For a: 4 = ( ) 4 =, since both and ( ) go to infinity. For b: 4 = ( ) 4 which DOES NOT EXIST, since here goes to infinity, and ( ) goes to. You cannot take the square root of a negative number, so the it doesn t eist. For c: +4 =, since the eponent is going to infinity. e Eercise : Compute the following its: Solution to #: a : For a: e5 = 0, since the eponent is going to. For b: ln( + 5) =. For c: ln( 1 ) = ln( 1 ) = ln() =. Rational Functions Eercise 4: Compute the following its: e5 b : ln( + 5) c : ln(1/) 4a : b : c :

2 Solution to #4: For each problem, we divide everything by the highest power of in the denominator. For 4a: For 4b: For 4c: = + 1 = = + 5 = = = + 5 = = Eercise 5: Compute the following its: 5a : b : c : Solution to #5: For each problem, we divide everything by the highest power of in the denominator. For 5a: For 5b: For 5c: = = 0 4 = = + = = = + = = 0 Other Fractions Eercise : Compute the following its. a : ln( ) b : e c : + Solution to #: In the following is a short for approaches or gets close to : means approaches, π means approaches π (from both sides), means approaches from the left, + means approaches from the right, etc. As usual, left side right side means that what s on the left lets you conclude what s on the right. Eample: z + = 15 z = 9 z =

3 Solution to a: + ln( ) 0 ln( ) = 0. Solution to b: e 0+ e e =. Solution to c: + + =. = (absolute value!) = Eercise : Compute the following its. a : b : c : Solution to a: = 5/ = 0; = = 0. Solution to b - makes use again of = : = = = ; = =. Solution to c: < 0 D.N.E.; D.N.E.. Horizontal Asymptotes Eercise 8 Find the horizontal asymptotes for each function on the worksheet. We shall it ourselves to the functions in the section Other Fractions. We remember the following generalities about horizontal asymptotes: A function f() has a horizontal asymptote on the left if f() eists as a (finite) real number l and f() has a horizontal asymptote on the right if + f() eists as a (finite) real number r (± are not permitted for either l or r ). We have two different horizontal asymptotes if both its eist and are the same (finite) real number. We have a single horizontal asymptote if only one of the its is a (finite) real number or if l = r.

4 Horizontal asymptotes for #4a: f() = ln( ) We have seen above that ln( = 0, the line y = 0 is a horizontal asymptote on the left. ) ( ) = f( ) = f() f() = f(). It follows that the line y = 0 also is a horizontal asymptote on the right, i.e., this is the only horizontal asymptote for f(). Horizontal asymptotes for #4b: f() = e We have seen above that f() blows up to as and there is no horizontal asymptote on the left. What about? Then e 0 e It follows that the line y = 0 is a horizontal asymptote on the right. e = 0. Horizontal asymptotes for #4c: f() = + We have seen above that =, the line y = is a horizontal asymptote on the left =, = (absolute value!) = + the line y = is a horizontal asymptote on the right. In particular, f() has two different horizontal asymptotes. Horizontal asymptotes for #5a: f() = We have seen above that = 0, the line y = 0 is a horizontal asymptote on the right Left hand side: < D.N.E.; the domain of f() does not etend all the way to the left and there is no horizontal asymptote on the left side. Horizontal asymptotes for #5b: f() = We have seen above that =, f() blows up and there is no horizontal asymptote on the left = = =, = = ; 4

5 same situation as on the left: f() blows up and there is no horizontal asymptote on the right either. There are no horizontal asymptotes for f(). Horizontal asymptotes for #5c: f() = We have seen above that D.N.E. and there is no horizontal asymptote on the left. = / =. This is not a finite it and there is no horizontal asymptote either. / = 14/ 9/ = 5/ ; In case you were wondering why 5/ = : You remember that for a base a > 1 the eponential function G() = a increases with. This means that a = a 1/ < a 5/ for all a > 1. Change of names: replace a with and you get < 5/ for all > 1 which certainly is the case if. But you know that as. This means that the bigger values 5/ also go to infinity. 5