Calculus/L'Hospital

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Expert: Alon Mandes - 1/11/2010


Question
hello
i've got :
1) Lim(x-Lnx) as x approaches infinity !! we should use the L.hospital Rule but it is ( infinity - infinity ) we should change it either to zero/zero or infinity/infinity but how do we do that ???


2)Lim((2x-3)/(2x+5))^(2x+1) as x approaches infinity
we use l'Hospital as well , but could you please show me how !!"

Answer
Hi Nora,
1) the function f(x)=x-ln(x) is rapidly increasing . Because always ln(x)<x , and when
  x is very big , ln(x) is much smaller that x . Therefore : Lim f(x) = infinity . No
  room for L.hospital Rule .
2) I will use he rule :
  Lim   (1+ a/x)^x = e^(a)  AND Lim   (1- a/x)^x = e^(-a)
  x->inf                        x->inf
  So,
  Lim ((2x-3)/(2x+5))^(2x+1) =
  x-> inf
  Lim ((2x+5-8)/(2x+5))^(2x+5-4) =
  x-> inf
  Lim (1- 8/A)^(A-4) =
  A-> inf
  Lim [(1- 8/A)^A ]*[1- 8/A]^(-4) = e^(-8)
  A-> inf
  Agian no room for L.hospital Rule .

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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