Verified answer

x/(x+5)

1-5/(x+5)

x^2/(x^2+x-20)-(4 x)/(x^2+x-20)

Root:

x = 0

Series expansion at x=0:

x/5-x^2/25+x^3/125-x^4/625+x^5/3125-x^6/15625+O(x^7)

Series expansion at x=∞:

1-5/x+25/x^2-125/x^3+625/x^4-3125/x^5+O((1/x)^6)

Derivative:

d/dx((x^2-4 x)/(x^2+x-20)) = (2 x-4)/(x^2+x-20)-((2 x+1) (x^2-4 x))/(x^2+x-20)^2

Indefinite integral:

integral (x^2-4 x)/(x^2+x-20) dx = x-5 log(x+5)+constant

RawBoxes[RowBox[{log, RowBox[{(, x, )}]}]] is the natural logarithm » | Documentation | Properties | Definition

Limit:

lim_(x->±infinity) (-4 x+x^2)/(-20+x+x^2) = 1

Series representation:

(-4 x+x^2)/(-20+x+x^2) = sum_(n=-infinity)^infinitypiecewise-(-1/5)^n | n>0\n0 | otherwise x^n