Home > Math Group > Question and Answer

Question:

Answers:

1. 
   jkj1362 Teacher
   

   pavelpimen,

   if f(x) = (1+x)^(1/x), limf(x) = e when x goes to infinity.

   'e' is a number that represents approximately about 2.718 

   Rate answer:

   • Flag as inappropriate

   Posted by jkj1362 on Friday May 22, 2009 at 2:15 AM
   
   

2. 
   neela Teacher
   Graduate School

   eNotes Editor

   We see that f(x)= (x+1)^(1/x) is a decreasing function as x becomes large.f(1)= 2,f(2)=3^(1/2) =1.732 nearly and f(10)=(11)^0.1 = 1.271  and f(1001)=(1001)^0.001 may be nearly 1 if you compute.

   Limit(x+1)(1/x) becomes like (inifinity)^0 , which is indetrminate form. If the limt exits , let it be L

   Taking logarithms, log L = limit (1/x)log(x+1), is in ifinity/infinity form.

   We use L'Hospital's rule to determine the limit, wherein we diferentiate both numerator and denominator and take limit.And if the indetermition persists, we use the technic again. We stop if there is no indetermination.

   Log L = [Limit(x->inf)(log x}'/ (x)'= [Limit (1/x)] /1=1/inf=0

   We conclude that log L = 0.

   Therefore, L = e^0 =1.

   Therefore, Limit x->inf (x+1)^(1/x) = 1

   Rate answer:

   • Flag as inappropriate

   Posted by neela on Friday June 5, 2009 at 2:00 AM