Determine whether the functions grow at the same rate or if one function grows faster than the other as $\displaystyle x \rightarrow \infty$

$\displaystyle 1)$

$\displaystyle lim_{x\rightarrow \infty} \frac{x\ln x}{x+\ln x}$

$\displaystyle lim_{x\rightarrow \infty} \frac{x\frac{1}{x} + \ln x (1)}{1+\frac{1}{x}} \rightarrow lim_{x\rightarrow \infty}\frac{1+\ln x}{\frac{x+1}{x}} \rightarrow lim_{x\rightarrow \infty} \frac{x +x \ln x}{x+1}$

$\displaystyle lim_{x\rightarrow \infty} \frac{1+ \ln x}{1}$

Since $\displaystyle x + \ln x$ reached 1 first, $\displaystyle x \ln x$ outgrew it. (This correct?)

$\displaystyle 2)$

$\displaystyle lim_{x\rightarrow \infty} \frac{(x^2 + 1)^{\frac{1}{2}}}{x}$

$\displaystyle lim_{x\rightarrow \infty}\frac{\frac{1}{2}(x^2+1)^{-\frac{1}{2}}(2x)}{1} \rightarrow lim_{x\rightarrow \infty} \frac{x(x^2+1)^{-\frac{1}{2}}}{1}$

$\displaystyle (x^2+1)^{\frac{1}{2}}$ outgrows $\displaystyle x$ because $\displaystyle x$ reached 1 first.

Was L' Hopitals rule supposed to be used in both of these? And how exactly do you know when to use the rule and when not to?