Calculate the limit of the function, if it exists? limit (x^2+x-6)/(x-2), x->2

We couldn't calculate the lim, by substituting x=2, because f(2) is not defined.

We cannot apply the Quotient Law, also, because the limit of denominator is 0, too.

We'll factorize the numerator, after finding it's roots

(x^2+x-6)/(x-2)

x^2+x-6=0

We'll use the quadratic formula:

x1=[-1+sqrt(1+24)]/2

x1=(-1+5)/2

x1=2

x2=(-1-5)/2

x2=-3

(x^2+x-6)/(x-2)=(x-2)(x+3)/(x-2)

After reducing the terms:

lim (x^2+x-6)/(x-2)=lim (x+3) = 2+3 =5