x趋于无穷大,lim[(x^2+x+1)/(x-1)-ax-b]=0,求a,b的值

Question

x趋于无穷大,lim[(x^2+x+1)/(x-1)-ax-b]=0,求a,b的值

Answer 1

lim(x→∞) [(x² + x + 1)/(x - 1) - ax - b] = 0
lim(x→∞) [(x² + x + 1) - (ax + b)(x - 1)]/(x - 1) = 0
lim(x→∞) [(x² + x + 1) - (ax² + bx - ax - b)]/(x - 1) = 0
lim(x→∞) [(1 - a)x² + (1 + a - b)x + (1 + b)]/(x - 1) = 0
{ 1 - a = 0 => a = 1
{ 1 + a - b = 0 => b = a + 1 = 2
∴a = 1,b = 2