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

Thread starter Zhalfirin88
Start date Feb 18, 2010
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Limit

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SUMMARY

The limit of the expression $\lim_{x\rightarrow 0} \left(\frac{5^x-1}{4x}\right)^{\frac{1}{x}}$ does not exist. As $x$ approaches 0 from the right, the limit approaches 0, while from the left, it approaches infinity. The discussion highlights the use of l'Hôpital's Rule to resolve the indeterminate form $0/0$ in the fraction $\frac{5^x-1}{4x}$, leading to the conclusion that the limit diverges based on the direction from which $x$ approaches 0.

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Understanding of limits in calculus
Familiarity with l'Hôpital's Rule
Knowledge of logarithmic properties
Basic understanding of exponential functions

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Study the application of l'Hôpital's Rule in limit problems
Learn about one-sided limits and their significance
Explore the behavior of exponential functions near zero
Investigate the concept of indeterminate forms in calculus

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Students studying calculus, particularly those tackling limits and indeterminate forms, as well as educators looking for examples of limit behavior in exponential functions.

Feb 19, 2010

Zhalfirin88

Class starts in 2 hours!

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