Does $\lim_{x\to 0^+} \sqrt{x}$ exist or not?

Question

Does this right-handed limit exist or not? $$\lim_{x\to 0^+} \sqrt{x}$$

Schaum's Easy Outline of Calculus (Second Edition) says it does. And doesn't.

An example in the book states:

The function $f(x)=\sqrt{x}$; then $f$ is defined only to the right of zero.

Okay. So the right-handed limit does exist.

Hence, $\lim_{x\to 0} \sqrt{x}=\lim_{x\to 0^+} \sqrt{x}=0$.

Okay. I'm still with you. The limit is $0$.

Of course, $\lim_{x\to 0^+} \sqrt{x}$ does not exist,...

Qué, Mr. Fawlty?

... since $\sqrt{x}$ is not defined when $x<0$.

Okay, so are they messing with me? Is my coffee too weak? Too strong? Is there some subtle truth about limits that escapes me?

Or is that a typo? Did they mean in that last line to omit the '$+$' by the '$0$' and write "Of course, $\lim_{x\to 0} \sqrt{x}$ does not exist,..."?

EDIT:
I think a minus sign is intended instead of a plus sign in that last limit.

Answer 1

Both $$ \lim_{x\to 0} \sqrt{x} \quad \text{and}\quad \lim_{x\to 0^+}\sqrt{x} $$ exist. In general for a function $f$ with domain $D(f)$, recall the definition of the $$ \lim_{x\to a} f(x) = L. $$ The definition says that this means that: For all $\epsilon >0$ there is a $\delta >0$ such that if $x\in D(f)$ and $0<\lvert x - a \rvert < \delta$ then $\lvert f(x) - L\rvert<\epsilon$. Often we don't write in the requirement that $x$ be in the domain of $f$, but this is a requirement.

Likewise the right hand limit exists.

See this Wikipedia article for more on this: https://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit#Precise_statement

Answer 2

Yes it exists, as you can check by $\varepsilon$-analysis.

Taking any $\varepsilon > 0$, we have $\sqrt{x} < \varepsilon$ if $0 < x < \varepsilon^{2}$.

Answer 3

Put a comma after "defined" in the last sentence. Or equivalently, brackets between "since" and "defined".

Shows why textbook authors really need good English as well as good maths! This happens an awful lot in academia. Very often when there's a bump, you're spending time solving a grammar problem! Understandable for research papers, but who killed off all the editors when it came to textbooks?