Is the following method correct? I was wondering if the following procedure is valid when calculating the limit: $$\lim_{x\to0}(\ln(x)\sqrt{1+x}-\ln(x))=\lim_{x\to0}(\ln(x)\sqrt{1}-\ln(x))=\lim_{x\to0}(\ln(\frac{x}{x})=0$$
Is this correct? If not, why?
 A: Your first equation is duious; in general, it's only the that
$$\lim_{x \to 0} \big( f(x)g(x) \big) = \left(\lim_{x \to 0} f(x)\right)\left(\lim_{x \to 0} g(x)\right)$$
if both $f(x)$ and $g(x)$ converge as $x \to 0$, but that is not the case here, since $\lim_{x \to 0} \ln(x) = -\infty$.
Here's another suggestion. Pulling out the common factor of $\ln(x)$ gives
$$\lim_{x \to 0} \ln(x)(\sqrt{1+x}-1)$$
Now $1=\sqrt{1}$, so the second term looks suspiciously like a limit definition of a derivative, namely the derivative of $\sqrt{x}$ at $x=1$. Multiplying by $\frac{x}{x}$ yields
$$\lim_{x \to 0} \left( x\ln(x) \cdot \frac{\sqrt{1+x}-\sqrt{1}}{x} \right)$$
But this is now the product of two functions which both converge as $x \to 0$, and hence you can split up the limits to obtain
$$\left( \lim_{x \to 0} x\ln(x)\right) \cdot \left( \lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1}}{x} \right) = \left(\lim_{x \to 0} x\ln(x)\right) \cdot \left[\frac{d}{dx}(\sqrt{x}) \right]_{x=1}$$
You can take it from here.
A: If your interest is more in the methodology rather than the actual limit, then you have stumbled upon a broader topic -- double limits.
In effect you are trying to evaluate this as a double limit -- which can be valid -- but not in this case.
What you have shown is one iterated limit
$$\lim_{x \to 0} \lim_{y \to 0} \ln x (\sqrt{1 +y} - 1) = \lim_{x \to 0}(\ln x \cdot 0 ) = 0.$$
By luck this coincides with the diagonal limit
$$\lim_{x \to 0} \ln x ( \sqrt{1 + x} - 1) = \lim_{x \to 0} \ln x ( 1 + x/2 + O(x^2) - 1) = 0.$$ 
However, with the order reversed we have
$$\lim_{y \to 0} \lim_{x \to 0} \ln x (\sqrt{1 +y} - 1) = \lim_{y \to 0} -\infty \cdot (\sqrt{1+y} - 1) = -\infty.$$
In general, when the iterated limits don't agree, then you can't be confidant that the diagonal limit coincides with one or the other.
If both iterated limits exist and one of the inner limits converges uniformly then we can be sure that
$$  \lim_{x \to 0} f(x,x) = \lim_{x \to 0} \lim_{y \to 0}f(x,y) = \lim_{y \to 0} \lim_{x \to 0}f(x,y).$$
