Question

is there a number a such that the lim x->2 6x^2+ax+a+57/x^+x-6 exists? if so find the value of a and the limit.

Answer 1

You have a fraction with denominator=0 at x=2

So, you want the numerator to also be zero, or you have no hope of having a limit.

6x^2 + ax + a + 57 = 0
24 + 2a + a + 57 = 0
81 + 3a = 0
a = -27

so,

6x^2 - 27x + 30 / x^2 + x - 6
=(6x-15)(x-2) / (x+3)(x-2)

So, the limit = (6x-15)/(x+3) = -3/5