Is it "safe" to evaluate $\lim_{x \to \infty} \frac{1}{\frac{1}{x} +x}$ as follows? If I want to evaluate the following limit:
$$\lim_{x \to \infty} \frac{1}{\frac{1}{x} +x},$$
is it valid to use regular arithemtic rules to come to
$$\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} \frac{1}{x} +\lim_{x \to \infty} x}?$$
Evaluating existing limits gives
$$\frac{1}{0 +\lim_{x \to \infty} x},$$
but writing something such as
$$\frac{1}{\infty}=0$$
does not seem legal, although it feels intuitive that the limit is $0$ from the first expression.
Is there a more sound way of stating this? Perhaps with different laws of limits which I seem to be lacking knowledge of? Thanks in advance for your help!

As explained in comments, your method is not valid because the hypotheses for the quotient and sum limit laws are not satisfied.  However, the common technique of (convert to polynomials, then) factor out the big part works.
\begin{align*}
\lim_{x \rightarrow \infty} \frac{1}{\frac{1}{x} + x}
    &= \lim_{x \rightarrow \infty} \left( \frac{1}{\frac{1}{x} + x} \cdot \frac{x}{x} \right)  \\
    &= \lim_{x \rightarrow \infty} \frac{x}{1 + x^2}  
\end{align*}
Here the largest power of $x$ appearing in the numerator or denominator is $2$, so factor out $x^2/x^2$.  Continuing the display,
\begin{align*} 
&=\lim_{x \rightarrow \infty} \left( \frac{x^2}{x^2} \cdot \frac{1/x}{1/x^2 + 1}  \right)  \\
&=\frac{\lim_{x \rightarrow \infty} 1/x}{\lim_{x \rightarrow \infty} (1/x^2 + 1)}  \\
&=\frac{\lim_{x \rightarrow \infty} 1/x}{\lim_{x \rightarrow \infty} 1/x^2 + \lim_{x \rightarrow \infty} 1}  \\
&= \frac{0}{0 + 1}  \\
&= 0
\end{align*}
This may seem like "put something in just so we can take it and a little bit more out again", and this is a valid observation.  Clearing the nested denominators is just to make it very clear where the power of $x$ is coming from.  However, we can do the same thing in the original form:
$$  \lim_{x \rightarrow 0}  \left( \frac{x}{x} \cdot \frac{1/x}{1/x^2 + 1} \right)  $$
equally arranging for the numerator and denominator to have finite limits with the denominator not having limit $0$.
