Limit of function. How can it suddenly change it's domain after simple manipulations I'm trying to refresh my math at the moment and have quickly become very confused by the calculation of limits of functions.
For example, I solved the following 
$\lim_{x \to 0} \frac{7x^2+4x^4}{3x^3-2x^2}$
by first manipulating it to 
$\frac{7+4x^2}{3x-2}$ and then concluding that the limit is $-\frac{7}{2}$
The thing I don't understand is why the original expression isn't defined in f(0) while the second one is? I'm not very experienced with math but I don't understand why the domain of the function can be changed by just multiplying the numerator and denominator of a fraction with the same value (which in this case is $\frac{1}{x^2}$).

You are dividing with $x^2$ which lies in the denominator and hence $x\neq0$. So it can't be $f(0)$ anymore.
