Find limit of polynomials Suppose we  want to  find limit of the following polynomial
$$\lim_{x\to-\infty}(x^4+x^5).$$
If we directly put here $-\infty$, we   get  "$-\infty +\infty$" which is definitely undefined form, but otherwise  if factor out $x^5$, our polynomial will be of the form $x^5(1/x+1)$.
$\lim_{x\to-\infty}\frac 1x=0$, so our result will be $-\infty*(0+1)$,which equal to $-\infty$. I have exam in a 3 days and interested if  my last procedure is correct? Directly putting $x$ values  gives me undefined form, but factorization  on the other hand, negative infinity, which one is correct?

Yes. generally if $\rm\:L\:$ is a field map preserving sums and products and $\rm\ L(g/f) = 0\ $ then
$$\rm L(f+g)\ =\ L(f\ (1+g/f))\ =\ L(f)\ \left(L(1) + L(g/f)\right)\ =\ L(f) $$
Induction yields the dominant term lemma: the L-value of a sum is that of the dominant term(s).
When $\rm\:L\:$ is a limit the map need not be total (everywhere defined). This concept is abstracted algebraically in the important notion of a valuation or place.
