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I have a question regarding $$\lim_{x\to 0} \sqrt{x}.$$

Is the limit $0$ or undefined? For the limit to exist both the right hand and left hand limits must exist and be equal. $\lim_{x\to 0^+} \sqrt{x} =0$ but it doesn't even make sense to talk about $\lim_{x\to 0^-} \sqrt{x}$ for the reals.
On the other hand If choose to apply the limit laws I get the following $$\begin{align} \lim_{x\to 0} \sqrt{x} & = \sqrt{\lim_{x\to 0} x}\\ & = \sqrt{0}\\ & =0. \end{align}$$

I am a bit confused and I need some clarification.

Thanks.

Gorg
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  • Definition of limit of a function $f$ at a point $a$ assumes that $f$ is defined in a deleted neighborhood of $a$. This simply means that $f$ must be defined in $(a - h, a + h)$ except possible at $a$ for some positive value of $h$. In this case $\sqrt{x}$ is not defined for any negative $x$ and hence we can't talk of its limit at $x = 0$. However we may talk about $\lim\limits_{x \to 0+}\sqrt{x}$ and this is $0$. – Paramanand Singh Aug 28 '13 at 09:15
  • Who is asking and why? – Stefan Smith Aug 29 '13 at 00:50

2 Answers2

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$\sqrt{x}$ is defined for all real $x$ when $x \lt 0$ the result is $i \sqrt{|x|}$ an imaginary number. Where $i = \sqrt{-1}$.

Consider

$$\lim_{x\to 0} \sqrt{x}$$

If we approach from above

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

Now if we approach from below

$$\lim_{x\to 0^-} \sqrt{x} = i \cdot \lim_{x\to 0^-} \sqrt{|x|} = i \cdot \lim_{x\to 0^+} \sqrt{x} = 0 \cdot i = 0 $$

The limit is defined from above and below and is the same in both cases so:

$$\lim_{x\to 0} \sqrt{x} = 0$$

Warren Hill
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We talk about right hand and left hand limits when there are points to the right and left hand sides within the domain of the function whose limit you are evaluating. In this case, there are no points in the domain of $\sqrt{x}$ on the left hand side of $0$. So, the only limit you can talk about is the right hand limit and it is $0$. So, the 'overall' limit is also $0$.

Parth Thakkar
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  • So when you say 'overall' limit, what you really mean is right hand limit? – Gorg Aug 28 '13 at 10:26
  • When we say that the limit is defined if left and right hand limits are defined and are equal, and in that case, then the limit is equal to the left and right hand limits. By 'overall' I meant, the limit as used here. I don't know if I am able to make it clear to you. – Parth Thakkar Aug 28 '13 at 17:51
  • Oh and btw, my argument is just considering real numbers - that's what I know. I do not know how calculus works in complex numbers. – Parth Thakkar Aug 28 '13 at 17:52
  • I think the use of left- and right-hand limits to define the "overall" limit is something you will see in some introductory (single-variable) calculus textbooks. It is what I recall as a high-school student. For multivariable calculus, "left" and "right" are no longer adequate to describe how a function approaches a limit. In order to define the limit of a function at the boundary of its domain, we might mention the domain in the definition of limit. So I think not every textbook will support this answer, but the more advanced ones will. – David K May 12 '23 at 13:31