Can't solve a difficult limit I need to solve this limit 
${\lim_ {x\to {+∞}}}{\frac{{x}(\sqrt{x^2 + x} - x) +\cos(x)\ln(x)}{\ln(1+\cosh(x))}}$
I've tried to use Taylor's Theorem with Peano's Form of Remainder, but first time I forgot that ${x\to{+∞}}$, so I made a substitution ${t=\frac{1}{x}}$, then I just didn't get anything (I've got ${o({\frac{1}{t}})}$ (or ${o((t-1)^3)}$ and too complicated expression) which doesn't disappear). I've thought to use L'Hospital's rule, but there's a problem with defining indeterminate form. Here we have ${\cos(x)\ln(x)}$ that sometimes becomes ${0\cdot∞}$. Then I thought about the existence of this limit and... WolframAlpha says it doesn't exist. But the answer in my book is 1/2.
So now Ii don't know how to solve it or does it even exist or not. Can anyone give me at least a hint of how to solve this problem?

Hint:
First the expression in two, and use equivalents:
We have $\cosh x\sim_{+\infty}\frac12\mathrm e^x$, so $1+\cosh x\sim \frac12\mathrm e^x$, and finally
$$\ln(1+\cosh x)\sim_{+\infty}x-\ln 2\sim_{+\infty} x$$
. On the other hand,
$$x(\sqrt{x^2 + x} - x)=\frac{x(\not x^2 + x - \not x^2)}{\sqrt{x^2 + x} + x}\sim_{+\infty}\frac{x^2}{2x}=\frac x2,$$
so that $$\frac{x(\sqrt{x^2 + x} - x)}{\ln(1+\cosh x)}\sim_{+\infty}\frac{\frac12 x}{x}=\frac 12.$$
Can you show that $$\frac{\cos x\ln x}{\ln(1+\cosh x)}\to 0?$$
