Is 1/(1+e^-(x)) equivalent to 1-e^-(x) when x is very large?

  • Context: Undergrad 
  • Thread starter Thread starter Cilly28
  • Start date Start date
  • Tags Tags
    Exponential
Click For Summary
SUMMARY

The expression 1/(1+e^-(x)) is approximately equal to 1-e^-(x) when x is greater than 50, particularly in the context of semiconductor hole density theorems. This approximation holds true as x approaches infinity, demonstrating that both expressions exhibit similar limiting behavior. The discussion emphasizes that the method of approximation should not invalidate the correctness of the result, especially when the underlying mathematical principles are sound. The author argues for the validity of their approach despite receiving an incorrect marking on their exam.

PREREQUISITES
  • Understanding of limits and asymptotic behavior in calculus
  • Familiarity with the exponential function and its properties
  • Knowledge of semiconductor physics, particularly hole density concepts
  • Experience with mathematical proofs and approximations
NEXT STEPS
  • Study the properties of the exponential function in detail
  • Learn about asymptotic analysis in calculus
  • Research semiconductor physics focusing on hole density and related theorems
  • Explore mathematical proof techniques and justification of approximations
USEFUL FOR

Students in physics or engineering, particularly those studying semiconductor theory, mathematicians interested in calculus, and educators assessing mathematical proofs and approximations.

Cilly28
Messages
1
Reaction score
0
Long story short, I said on an exam that...

1/(1+e^-(x)) =~(is nearly equal to) 1-e^-(x); note* x > 50 , they are also inversely proportional when x<-50 but that isn't specific to the problem

I used this to prove a theorem regarding hole density in semiconductors ""when x is very very large""...It was marked incorrectly but the question stated that the magnitude of x would be >50. I noticed the limiting behavior of this as it goes from 50 to infinity and they appear to be equal, especially when approximations are made in the 'books proof' to simplify the equation, mainly throwing low integers away that do not effect a number of much higher magnitude.

If I made a correct approximation and still obtained the same result equation, just because I did it differently shouldn't make it incorrect, would anyone agree?
 
Physics news on Phys.org
The apporximation is correct. I suggest you ask the person who marked it incorrect explain why. It may be that you were supposed to justify the approximation in more detail.
 

Similar threads

High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
Find the range of the function ##f(x) = \frac{e^{2x}-e^x+1} {e^{2x}+e^x+1}##
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Why is [x^x^x]' NOT (xln(x)^2+ln(x)+xln(x)+1)*x^(x+x^x)?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Why the integral of a complex exponential can't be equal to zero?
  • · Replies 6 ·
Replies
6
Views
3K
High School Mass estimate only through mass rate of change
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Intersections between this infinite power tower and a multifactorial
  • · Replies 1 ·
Replies
1
Views
2K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Derivative of a^x: Learn to Calculate M(a) and e^x
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Invert a 3D Fourier transform when dealing with 4-vectors
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K