Limit of product of mixed root indexes I am unable to recover the limit of this function.
$$
\lim_{x \to +\infty} \left (\frac{1}{\sqrt{x}} \times  \sqrt[3]{x+1}  \right )
$$
I have tried many ways of solving it. These are the two simplest:


*

*Variable substitution
Using $y = \sqrt[3]{x+1} $, with which I got a common factor $\sqrt{y-1}$ and got
$$
\lim_{x \to +\infty} \left (\frac{1}{\sqrt{y^3-1}} \times  y  \right) = 
\lim_{x \to +\infty} \left (\frac{\sqrt{y^2}}{\sqrt{y^3-1}} \right) =
\lim_{x \to +\infty} \left (\frac{\sqrt{y^2}}{\sqrt{y^3(1-\frac{1}{y^3})}} \right) =
\lim_{x \to +\infty} \left (\frac{1}{\sqrt{y \left (1-\frac{1}{y^3} \right)}} \right) = \frac{1}{+\infty \times 1} = 0
 $$


*

*Using the common factor $\sqrt{x}$
$\sqrt[3]{x+1} = \sqrt[6]{(x+1)^2} = \sqrt[6]{x^2+2x+1} = \sqrt[6]{x^3 (\frac{1}{x}+\frac{2}{x^2}+\frac{1}{x^3})}=\sqrt{x} \left  (\sqrt[6]{\frac{1}{x}+\frac{2}{x^2}+\frac{1}{x^3}}  \right )$
and therefore I got 
$$
\lim_{x \to +\infty} \left (\frac{\sqrt{x} \left  (\sqrt[6]{\frac{1}{x}+\frac{2}{x^2}+\frac{1}{x^3}}  \right )}{\sqrt{x}} \right ) =
\lim_{x \to +\infty} \left (\sqrt[6]{\frac{1}{x}+\frac{2}{x^2}+\frac{1}{x^3}} \right ) = \sqrt[6]{\frac{1}{+\infty}+\frac{1}{+\infty}+\frac{1}{+\infty}} = 0
$$
However, I found that the limit is positive when I plotted a function with this expression. I have plotted it here, you can look at it yourselves. This means my calculations are wrong. What am I doing wrong and how do I solve it correcly?
Thank you for the help.

Your calculations are right!
$$
\begin{aligned}
\lim _{x\to \infty }\left(\frac{\sqrt[3]{1+x}}{\sqrt{x}}\right)
& = \lim _{t\to 0\:}\left(\frac{\sqrt[3]{1+\frac{1}{t}}}{\sqrt{\frac{1}{t}}}\right)
\\& = \lim _{t\to 0\:}\left(\sqrt[6]{t}\sqrt[3]{t+1}\right)
\\& =0^{\frac{1}{6}}\sqrt[3]{0+1} = \color{red}{0}
\end{aligned}
$$

