Originally Posted by
HallsofIvy This is the same as
$\displaystyle \lim_{\Delta x\to 0}\frac{6x^2\Delta x+ 6x (/Delta x)^2+ 6x\Delta x+ 3(\Delta x)^3}{(4x^2+ 8x\Delta x+ 4x+ 1)\Delta x}$
Since there is a "$\displaystyle \Delta x$" in each term in the numerator, you can cancel that "$\displaystyle \Delta x$" multiplying the denominator:
$\displaystyle \lim_{\Delta x\to 0}\frac{6x^2+ 6x\Delta x+ 6x+ 3(\Delta x)^3}{4x^2+ 8x\Delta x+ 4x+ 1}$
Now, neither numerator nor denominator goes to 0 as $\displaystyle \Delta x$ goes to 0.