The answer seems ok to me, in that it shows that the student understands what the limiting behavior of $x - \sqrt{x}$ is as $x \to \infty$. If the questioner wants to see a formal justification of that, then the word "find" should be replaced by some more precise indication.
The interpretive problem is that that the exercise could be better posed. A better way to ask the same question is simply to ask: what is the limiting behavior of $x - \sqrt{x}$ as $x \to \infty$?
In the classical sense the limit $\lim_{x \to \infty}x^{2}$ does not exist. What is meant by writing $\lim_{x \to \infty}x^{2} = \infty$ is different than what is meant by $\lim_{x \to 2}x^{2} = 4$, because $\infty$ is not a real number. The expression $\lim_{x \to \infty}f(x) = \infty$ is a shorthand that means that $f(x)$ is not bounded on the positive real line. It is used to distinguish the behavior of $\log{x}$ and $\cos{x}$ as $x \to \infty$. Neither has a limit as $x \to \infty$ in the usual sense, but their limiting behaviors are different, in the sense that one function grows unboundedly while the other is oscillatory, although bounded. This can be indicated by assigning the putative "value" $\infty$ to $\lim_{x \to \infty}\log{x}$ and declaring that $\lim_{x \to \infty} \cos{x}$ "does not exist", even though $\infty$ to $\lim_{x \to \infty}\log{x}$ also does not exist in the usual sense (it cannot be assigned a value in $\mathbb{R}$). So one is using the same notation to indicate a limit that exists in the usual sense and a limit that, although it does not exist in the usual sense, can be given sense in that the limiting behavior of the argument function has a well-defined character ... This is akin to the unfortunate and sloppy conflation of integrals and primitives that one often encounters in first calculus courses; apparently identical notation is used to indicate operationally and/or conceptually distinct entities.
Because the relevant distinction is a subtle for students, it seems to me a pedagogical error to treat $\lim_{x \to \infty}x^2 = \infty$ and $\lim_{x \to 2}x^{2} = 4$ on the same level. Their meanings and interpretations are different. I would try to avoid writing the former expression, or would make a serious effort to explain that it is formal notation indicating something different than the well-defined notation in the latter expression (one encounters similar issues when treating integrals with infinite limits or integrals of unbounded functions).
When one asks "find $\lim_{x\to \infty}f(x)$" it is implicit in the question being well posed that $f(x)$ has a well-defined limiting behavior that can be summarized by a single notational expression (so is not oscillatory), but "find" is really the wrong word.