[Calculus: Limits] How is this problem solved?

[u/synthsync_]

[u/Dry-Fondant-3614]

a^x = e^xlna

LHospitals rules says if limit is both 0 or both infinite in numerator and denominator the limit of the derivative of both the numerator and denominator is the same as the limit

Derivative of x = 1

Derivative of e^xlna is lnae^xlna which tends to infinity

[u/synthsync_ → u/Dry-Fondant-3614]

I’m confused, my professor told us that it’s a proof that we must memorise. Is it a proof?

[u/Dry-Fondant-3614 → u/Dry-Fondant-3614]

Oh I just saw the without LHospitals rule ...

Well the exponential with increase far faster than x at infinity.

Very easy to show with Taylor series

The -1/x goes to zero ignore it.

You have a^x / x

Just show that for all x' > 1 and x' > x

a^x /x < a^x' / x'

So the function must increase indefinitely this tends to infinity.

[u/CyclingMack → u/Dry-Fondant-3614]

L’Hopital. The hospital of limits is wonderful.

[u/Mr_Biscuits420]

exponential functions are the fastest rising functions, while linear are mostly slow

a^x - exponential , and thus it should be infinity

[u/CyclingMack → u/Mr_Biscuits420]

Mr Biscuits. I think you had a good idea. Let’s make language a bit different. Exponential functions with base greater than one will have a greater rate of increase than linear functions as x increases.

[u/Davidiying → u/Mr_Biscuits420]

a^x is bigger than x^a which is bigger than log(x)

[u/MathFunky]

Let's dissect this function f(x)=(a^x-1)/x into a^x/x-1/x.
Therefore limx->∞f(x)=limx->∞ a^x/x- limx->∞ 1/x.

1/x is negligibly tiny.

So it's essentially limx->∞ a^x/x. a^x is exponential, because a>1. And exponential functions are higher than the exponent. So as x goes to infinity, a^x goes way above infinity. this makes the answer ∞.

[u/payinthefidlr]

For a more rigorous approach than has been suggested, you can calculate the taylor expansion at x=0 of the numerator, which allows you to cancel the x in the denominator. This will leave you with an infinite polynomial where all of the coefficients are positive, and thus the limit diverges towards positive infinity as x goes to infinity

[u/Nabstar333]

Irrelevant comment: I thought it was L'Hopital lol. He sounds french but Im not sure.