posted by Blue on .
Find the constant c so that
lim [x^2 + x + c]/[x^2 - 5x + 6] exists.
For that value of c, determine the limit. (Hint: Find the value of c for which x - 3 is a factor of the numerator.)
factor the bottom
factor the top with (x-3) a factor so we can cancel it
(x-3)(x+b) = x^2+x+c
x^2 - 3x +b x - 3 b = x^2 + x + c
b x -3 x = x so b = 4
then c = -12
(x^2 + x - 12)/(x^2 - 5 x + 6)
(x-3)(x+4) / [(x-3)(x-2)]
when x = 3
7/1 = 7
hmmm. looks familiar. Anyway,
denominator is (x-3)(x-2), so we want the top to be
(x-3)(x-k) so that the fraction exists everywhere except at x=3, but has a finite limit there.
x^2 - (k+3)x + 3k = x^2 + x + c
k+3 = -1
k = -4
so c = 3k = -12
x^2 + x + c = x^2 + x -12 = (x-3)(x+4)