Find the limit of ((x^3+x+3)^(1/2)-x)^x as x approaches infinity

 Exercise

$\lim_{x\to\infty}\left(\sqrt{x^3+x+3}-x\right)^x$

 

Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$

$\lim_{x\to\infty }\left(e^{x\ln\left(\sqrt{x^3+x+3}-x\right)}\right)$

 

Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$

${\left(\lim_{x\to\infty }\left(e\right)\right)}^{\lim_{x\to\infty }\left(x\ln\left(\sqrt{x^3+x+3}-x\right)\right)}$

 

The limit of a constant is just the constant

$e^{\lim_{x\to\infty }\left(x\ln\left(\sqrt{x^3+x+3}-x\right)\right)}$

 

Evaluate the limit $\lim_{x\to\infty }\left(x\ln\left(\sqrt{x^3+x+3}-x\right)\right)$ by replacing all occurrences of $x$ by $\infty $

indeterminate

indeterminate

 Try other ways to solve this exercise

Can't find a method? Tell us so we can add it.