Can the difference of 2 undefined limits be defined? Is this limit defined or undefined?
$$\lim\limits_{x \to 0+} \left(\sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{x}}\right)$$
When I apply the rule of difference of limits, it's undefined. But, when I manipulate it, it gives me zero. And the graph of the function indicates it's defined on the right side.

By multiplying by $\frac{\sqrt{\frac{1}{x}+2}+\sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+2}+\sqrt{\frac{1}{x}}}$:
$$\lim\limits_{x \to 0+} \frac{\left( \sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{x}} \, \right) \left(\sqrt{\frac{1}{x}+2}+\sqrt{\frac{1}{x}} \, \right)}{\sqrt{\frac{1}{x}+2}+\sqrt{\frac{1}{x}}}$$

$$=\lim\limits_{x \to 0+} \frac{\frac{1}{x}+2-\frac{1}{x}}{\sqrt{\frac{1}{x}+2}+\sqrt{\frac{1}{x}}}$$
$$=\lim\limits_{x \to 0+} \frac{2}{\sqrt{\frac{1}{x}+2}+\sqrt{\frac{1}{x}}}$$
Then, we multiply by $\frac{\sqrt{x}}{\sqrt{x}}$:
$$=\lim\limits_{x \to 0} \frac{2\sqrt{x}}{\sqrt{1+2x}+1}$$
And, we substitute:
$$=\frac{2\sqrt{0}}{\sqrt{1+2\times0}+1} = 0$$
So, is this limit defined or not? and what's my error, if any?
Remember that the rule that you referred to, "the rule of difference of limits", is not just the equation 
$$
\lim_{x\to a}(f(x)-g(x))=\lim_{x\to a}f(x)-\lim_{x\to a}g(x)
$$
but rather the statement that, if both of the limits on the right side of this equation are real numbers, then the limit on the left side (is also a real number and) is given by this equation. So this rule does not apply to the limit in your question.

More generally, when learning rules (or theorems or principles or whatever they may be called), don't just learn formulas, but pay attention also to the words around them. The words are not just decoration but are essential for the correctness of the rule.
What you wrote,$$\lim\limits_{x \to 0+} \sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{x}},$$is undefined, because it is the difference of$$\lim\limits_{x \to 0+} \sqrt{\frac{1}{x}+2},$$which is undefined, and a another quantity$$\sqrt{\frac{1}{x}},$$the latter being meaningful, although dependent on the variable $x$. What I think you meant to write is $$\lim\limits_{x \to 0+}\left( \sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{x}}\right).$$ This is not, as the title of your question says, "the difference of two undefined limits". Rather, it is a (single) limit of a quantity (that happens to be expressed as a difference). As long as you treat this as a single limit, which is what you have done quite correctly, no problem arises.

In general, it is a false move to split a limit of a difference into the difference of two limits. It works sometimes, but then it needs to be justified carefully. On the other hand, if two limits are separately well defined, then their difference can be expressed as the limit of a difference (assuming that the variable, domain, and endpoint of the limiting process is the same for both).