# Application of L'Hopital's Rule

Task:

Graph the f(x) = e^2x - 1/x

Verify the Limit x→0 f(x) meets the criteria for applying L'Hopital's Rule

Find the Limit x→0 f(x)

Explain why L'Hopital's Rule cannot be used to find the limit of Lim x→0 e^2x/x

https://brainmass.com/math/calculus-and-analysis/application-of-l-hopital-s-rule-570901

#### Solution Preview

for f(x) = (e^2x - 1)/x, if x goes to 0, then both e^2x - 1 and x go to 0. That is, ...

#### Solution Summary

The solution explains when L'Hopital's Rule works and how it works with examples.

Differentiation

Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial theorem or equivalently using induction on n. Feel free to use this result although anything else should be made rigorous. It should be possible to prove this by the basic definition of the derivative. Thanks! Please no fancy inverse function theorem, just the basic definition of the derivative...

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