How to compute $\lim\limits_{x \to +\infty} \left(\frac{\left((x-1)^2-x\ln(x)\right)(x)!}{(x+2)!+7^x}\right)$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it.
Can you explain the method and the steps used?
$$\lim\limits_{x \to +\infty} \left(\frac{\left((x-1)^2-x\ln(x)\right)(x)!}{(x+2)!+7^x}\right)$$

Just look at this part of it: 
$\frac{(x-1)^2 x!}{(x+2)!}$ these are the largest terms top and bottom. This has limit $1$. It should now be clear how to do a rigorous proof, for example multiply top and bottom by $\frac{1}{(x+2)!}$ then all of the terms will have finite limits.
