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: Note that, for large $x$, $\cosh x$ behaves as $\frac12e^x$. So, $\lim_{x\to\infty}\frac{\log(\cosh x)}x=1$. And your limit is equal to$$\lim_{x\to\infty}\frac{\left(\sqrt{x^2+x}-x\right)+\frac{\cos(x)\log(x)}x}{\frac{\log(\cosh x)}x}.$$
