# Math Help - How do I get(prove/solve) these limits? "lim x->+infinity (x)^(2/ln(1+x^3))" ...?

1. ## How do I get(prove/solve) these limits? "lim x->+infinity (x)^(2/ln(1+x^3))" ...?

Hello, i've got some questions i really don't know how to prove/solve, and thats limits of infinity involving fraction as exponents.(lim x->+infinity (x)^(2/ln(1+x^3)) ) .

here are some questions that puzzles me on:

pi = the "pi" sign
sqrt. of = sqrt. of a number
infinity = infinity sign

1.) lim x->+infinity (x)^(2/ln(1+x^3))
2.) lim x->0^+ (cos sqrt of x)^(1/x)
3.) limx->1-0 ((2/pi)(arcsin(x)))^(1/acrcos(x))
4.) lim x->0^+ (x)^((1)/(1+ln sqrt. of x))
5.) lim x->+infinity (2+e^x)^(-1/x)
6.) lim x->0 (1+3x)^(2/sin(x))
7.) lim x->0 ((a^x+b^x+c^x)/(3))^(1/x)

well, I've already encounter and solve some limits but not related on these parts(fractions as exponents)
tried reading my book but didn't show some examples on how to solve it.

If you know how to solve it, please let me know.
or if you've got some examples,then show it to me.
And

Hello, i've got some questions i really don't know how to prove/solve, and thats limits of infinity involving fraction as exponents.(lim x->+infinity (x)^(2/ln(1+x^3)) ) .

here are some questions that puzzles me on:

pi = the "pi" sign
sqrt. of = sqrt. of a number
infinity = infinity sign

1.) lim x->+infinity (x)^(2/ln(1+x^3))
2.) lim x->0^+ (cos sqrt of x)^(1/x)
3.) limx->1-0 ((2/pi)(arcsin(x)))^(1/acrcos(x))
4.) lim x->0^+ (x)^((1)/(1+ln sqrt. of x))
5.) lim x->+infinity (2+e^x)^(-1/x)
6.) lim x->0 (1+3x)^(2/sin(x))
7.) lim x->0 ((a^x+b^x+c^x)/(3))^(1/x)

well, I've already encounter and solve some limits but not related on these parts(fractions as exponents)
tried reading my book but didn't show some examples on how to solve it.

If you know how to solve it, please let me know.
or if you've got some examples,then show it to me.
And

Have you studied L'Hospital rule?

$\displaystyle{\lim\limits_{x\rightarrow\infty}\fra c{\ln x}{\ln(1+x^3)}=\lim\limits_{x\rightarrow\infty}\fr ac{x^3+1}{3x^3}=\frac{1}{3}}$ , applying L'H rule, so

$\displaystyle{x^{\frac{2}{\ln(1+x^3)}}=e^{2\frac{\ ln x}{\ln(1+x^3)}}\xrightarrow [x\to\infty]{}e^{2/3}}$ .

Without L'Hospital I don't know how to evaluate the above limit, and most of the other ones you wrote.

Tonio

3. Originally Posted by tonio
Have you studied L'Hospital rule?

$\displaystyle{\lim\limits_{x\rightarrow\infty}\fra c{\ln x}{\ln(1+x^3)}=\lim\limits_{x\rightarrow\infty}\fr ac{x^3+1}{3x^3}=\frac{1}{3}}$ , applying L'H rule, so

$\displaystyle{x^{\frac{2}{\ln(1+x^3)}}=e^{2\frac{\ ln x}{\ln(1+x^3)}}\xrightarrow [x\to\infty]{}e^{2/3}}$ .

Without L'Hospital I don't know how to evaluate the above limit, and most of the other ones you wrote.

Tonio
we have discuss it last time but the teacher didn't give at least one sample on solving it(and i didn't quite understand it yet).
anyway mind to check if my answers for no.2,4, and 5 are correct
in no. 2 I get an answer of -0.5 ; no.4 e^2 ; no.5 -1
while i'm still tryingto solve 3,6,7

The answer in 2 is $e^{-1/2}$ , not -1/2 , 4 is correct and you don't even need limits for that but only simple
properties of logarithms, and the answer in 5 is $e^{-1}$ , not -1...you seem to be forgetting the exponential function.