Question

Evaluate the limit using L'Hospital's rule if necessary

Answer 1

\[\lim_{x \rightarrow + \infty} x^{x^8/x}\]

Answer 2

\[\LARGE \lim_{x\rightarrow \infty}x^{\frac{x^3}{x}}\]
This?

Answer 3

No, its 8 in the top. not 3

Answer 4

x^(8/x)

Answer 5

\[\Large \lim_{x\rightarrow\infty}x^{\frac{8}{x}}\]

Answer 6

yes/

Answer 7

here is a question
what is the meaning of \(\large x^{\frac{8}{x}}\) ?

Answer 8

exponential numbers that goes 0?

Answer 9

answer:
\[x^{\frac{8}{x}}=\exp(\frac{8\ln(x)}{x})\]

Answer 10

exp stands for exponential right?

Answer 11

yes

Answer 12

how do you get specific number?

Answer 13

instad of fruction

Answer 14

you mean for the limit?

Answer 15

\[\large f(x)=x^{\frac{8}{x}}=\exp(\frac{8}{x}\ln(x))\]

Answer 16

if you want to compute a number, then you compute
if you want to compute
\[\lim_{x\to \infty}f(x)\] then since the exponential is a continuous function
you compute the limit inside, i.e. compute
\[\lim_{x\to \infty}\frac{8\ln(x)}{x}\]

Answer 17

i am not sure that limit is obvious or not, if not, you can use l'hopital

Answer 18

it may be clear that the limit is zero, because \(x\) grows much faster than \(\ln(x)\)
if that is not obvious, then you can do more work
since the limit is zero you get
\[\exp(0)=1\] as your answer

Answer 19

you may have been taught "take the log, take the limit, then exponentiate" and that works, but it ignores the real meaning of an exponential function

Answer 20

O.. I see. exponential and log always get me confused!

Answer 21

\[x^{\frac{8}{x}}\] take the log get
\[\ln(x^{\frac{8}{x}})=\frac{8}{x}\ln(x)\] etc

Answer 22

then since step one was to take the log, after you find the limit is zero you get your answer as \(e^0=1\)

Answer 23

Thank you so much!!