An Infinite Limit?

In your very first step, you cannot break a limit into the subtraction of two limits unless both of the other limits exist. The theorem:

$\lim _{x\to a} (f(x)+g(x))=\lim _{x\to a}f(x) +\lim _{x\to a}g(x)$ is ONLY valid if the two limits on the right hand side exist. In your case, the second limit clearly does not exist, because it goes to infinity.

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Edit for clarity, neither does the first limit. So in effect, what you tried to do was make this an $\infty - \infty$, which doesn't work as seperate limits, but does work together (sometimes)