factored form of polynomial

[martin117]

how they replaced lim x tends to a g(x)=g(a) to g(x)=(x-a)^k g(sub)1 (x)
and is this also equal to g(a) . And same for h(x).

 

[tkhunny]

This is a matter of counting.
1) The fact that you found a factor of (x-a) in the denominator says x = a is NOT in the Domain. No matter what else happens, this won't change.
2) You found common factors of (x-a) in numerator add denominator. This just means there is some thinking to do.
3) Away from x = a, it is always the case that (x-a)/(x-a) = 1. This process of eliminating common factors by substituting "1" for each pair is usually called "cancelling" or something equally confusing.
4) Who runs out first?
----- 4a) Numerator runs out of (x-a) factors first, leaving some (x-a) factor(s) in the denominator, the limit fails to exist. We usually call this a Vertical Asymptote. The degree of the remaining denominator factors of (x-a) gives information about the behavior of the vertical asymptote
--------- 4a1) Even degree - both sides go off in the same direction (both up or both down).
--------- 4a2) Odd Degree - sides shoot off in opposite directions (one up and on down).
----- 4b) Both run out of (x-a) factors at the same time - the limit exists. We usually call this a "hole".
----- 4c) Denominator runs out of (x-a) factors first... See 4b).