What is the limit of (a^x - 1)/x as x approaches 0?

  • Thread starter Thread starter wiz0r
  • Start date Start date
  • Tags Tags
    Limit
Click For Summary

Homework Help Overview

The discussion revolves around evaluating the limit of the expression (a^x - 1)/x as x approaches 0, specifically exploring why the limit equals log(a).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of L'Hospital's rule and its application to the limit. There is also mention of using the Maclaurin series for a^x as an alternative approach. Questions arise regarding the reasoning behind the limit's value and the possibility of solving it without L'Hospital's rule.

Discussion Status

The discussion is active, with participants exploring different methods to evaluate the limit. Some guidance has been offered regarding the use of derivatives and series expansion, but there is no explicit consensus on the preferred approach yet.

Contextual Notes

Participants are navigating assumptions about the limit and the mathematical tools available to them, such as L'Hospital's rule and series expansions, without resolving the underlying concepts fully.

wiz0r
Messages
56
Reaction score
0

Homework Statement



lim x->0 ( (a^x - 1)/x )

Homework Equations



NA

The Attempt at a Solution



The professor told me that the answer to that limit is log(a), but why? I don't understand; can someone explain why?
 
Physics news on Phys.org
Do you know l'hopital's rule?
 
gb7nash said:
Do you know l'hopital's rule?


Yeah, I just figured it out. Take the derivative of both the numerator and the denominator and then it's easy from there...

Is there any way to do it without having to use L'Hospital's?
 
You could take the maclaurin series of a^x and plug it in for a^x. Once you do this, stuff cancels out and you'll obtain the same answer.
 

Similar threads

Limit of 0^0: Evaluating x^sinx
  • · Replies 19 ·
Replies
19
Views
2K
Investigating Limits: What Happens When x Approaches a Constant or Zero?
  • · Replies 1 ·
Replies
1
Views
1K
Calculate the limit cos(x)/sin(x) when x approaches 0
  • · Replies 12 ·
Replies
12
Views
4K
Limit and Integration of ##f_n (x)##
  • · Replies 8 ·
Replies
8
Views
2K
Use the graph of f(x) to investigate the limit
  • · Replies 6 ·
Replies
6
Views
2K
Proving Limit as x Approaches 0 for f(x)/x is 1?
  • · Replies 7 ·
Replies
7
Views
2K
Rules to apply L'Hospital on a limit
  • · Replies 12 ·
Replies
12
Views
2K
The integral of (sin x + arctan x)/x^2 diverges over (0,∞)
  • · Replies 5 ·
Replies
5
Views
3K
What is true for limit of f (x,y) as (x,y)→(0,1)?
  • · Replies 3 ·
Replies
3
Views
2K
Find limit of multi variable function
  • · Replies 10 ·
Replies
10
Views
2K