Finding the Limit of $\frac{5^x-1}{4x}^\frac{1}{x}$

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Homework Help Overview

The problem involves finding the limit of the expression \(\lim_{x\rightarrow 0} \left(\frac{5^x-1}{4x}\right)^{\frac{1}{x}}\), which falls under the subject area of limits and exponential functions.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss taking the natural logarithm of the limit expression to simplify the evaluation. There are attempts to analyze the behavior of the fraction \(\frac{5^x-1}{4x}\) as \(x\) approaches 0, with some noting it leads to an indeterminate form. Others suggest using l'Hôpital's rule to resolve the indeterminate form.

Discussion Status

The discussion is ongoing, with various participants exploring different interpretations of the limit and the implications of approaching from the left or right. Some guidance has been offered regarding the use of logarithms and l'Hôpital's rule, but there is no explicit consensus on the limit's value.

Contextual Notes

There is mention of the limit potentially being different when approaching from positive versus negative values of \(x\), indicating a need to clarify the direction of the limit. Participants express uncertainty about the nature of the limit and its existence.

  • #31
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So the answer is that the limit does not exist as x approaches zero? And then explain the right and the left limits? I should have realized that sooner because I even typed it into a graphing calculator -.-

Btw, this was a bonus question, they are harder and/or trickier :D
 

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