Question

HELP PLEASE ASAP CALCULUS !!

Answer 1

@theDeviliscoming @Nnesha @Bob please help :-)

Answer 2

if we plug in x = infinity we get infinity / infinitty

we could try l'hopitals rule
first derivative = 20x^19 / e^x
if we keep on differentiating we would get 20! in the numerator and e^(20!) as denominator
and e^(20!) will be a lot greater
so i would say the limit is 0.

Answer 3

also positive power functions grow much slowly than exponential functions so at the limit the fraction x^20 / e^x will tend to zero.

Answer 4

I'm sure of the answer but I'm not sure if l'hopital's rule applies here

Answer 5

https://www.desmos.com/calculator/wqkwk31x8q

theres the graph of the function the values drops steeply at about x = 100

Answer 6

both of the cat's suggestions are good.

the 2nd one is the best, ie exponential growth > power of 20 growth, but requires a bit of experience i think.

L'Hopital: yes it applies because this is in \(\dfrac{\infty}{\infty}\) indeterminate form


so,.... you could show that In general:

\(\dfrac{d^k}{dx^k} \left( x^n \right) = \dfrac {n!}{(n-k)!} x^{n-k} \)

....or take the first 20 derivatives of each..... :)

you could also use the power series deinition of \(e^z\):

\( e^z=\sum\limits _{k=0}^{\infty }{z^{k} \over k!}=1+z+{z^{2} \over 2}+{z^{3} \over 6}+{z^{4} \over 24}+\cdots \)

Then some algebra...easier than differentiating 20 times.

But cat's 2nd argument is the best i can think of FWIW ATM

Answer 7

\(\displaystyle\frac{x}{e^{x/20}}\to0\text{ as }x\to\infty\) (you can use L'Hospital once).

Then \(\displaystyle \frac{x^{20}}{e^x}=\left(\frac{x}{e^{x/20}}\right)^{20}\)