Ok, so I almost got this one. It just doesn't match the answer at the back of the book and I keep checking to see where I made a mistake but I can't quite find it. Here's what I did for the initial setup:

$\displaystyle\lim_{\Delta x \to 0} \frac{(x+\Delta x)^2-\frac{1}{x+\Delta x}-x^2+\frac{1}{x}}{\Delta x}$

And then skipping a few steps (getting a common denominator $x(x+\Delta x)$, moving the $\Delta x$ into the upper denominator, and then getting rid of terms that cancel):

$\displaystyle\lim_{\Delta x \to 0} \frac{2x^3\Delta x+3x^2\Delta x^2+x\Delta x^3+\Delta x}{\Delta x(x)(x+\Delta x)}$

And then the $\Delta x$ in the denominator would take care of many of them in the numerator…

$\displaystyle\lim_{\Delta x \to 0} \frac{2x^3+3x^2\Delta x+x\Delta x^2+1}{(x)(x+\Delta x)}$

And finally, putting 0 in for $\Delta x$ would give us:

$\displaystyle\lim_{\Delta x \to 0} \frac{2x^3+1}{x^2}$

Even though the answer in the back of the book is:

$\displaystyle\lim_{\Delta x \to 0} \frac{2x+1}{x^2}$

If I need to give more info on the steps I skipped, I will. Thanks!