Find the values a and b

Question

find the values of a and b

lim x→2 x²+ax+b/x³-2x²+x-2 = 6

Answer 1

Let us write the function in the limit as f(x)/g(x), and further factor f(x)=x²+ax+b=(x-r)(x-s) and g(x)=x³-2x²+x-2=(x-2)(x²+1). Since g(2)=0, if f(2)≠0, then the limit would not exist. Therefore, 2 is a root of f(x), so without loss of generality we can set r=2. Away from x=2, the factors of (x-2) in f(x) and g(x) cancel, and we can write f(x)/g(x)=(x-s)/(x²+1). The limit as we approach x=2 is therefore (2-s)/5=6. Solving for s, we get s=-28, so f(x)=(x-2)(x+28)=x²+26x-56, so a=26, b=-56.