Evaluate the limit

Question

Lim_x→-2 ((2/3)+(4/x-4))/(x+2)

Answer 1

f'(c) = lim_x→c {[f(x) - f(c)] / (x-c)}

 

Let f(x) = 4/(x-4).  Then f(-2) = 4/(-6) = -2/3 and f'(x) = -4 / (x-4)²

 

So, the given limit is equivalent to lim_x→-2 {[ -f(-2) + f(x)] / (x - (-2)}

 

                                              = lim_x→-2{(f(x) - f(-2)) / (x-(-2))}

 

                                              = f'(-2)

 

                                              = -4 / (-2-4)² = -4 / 36 = -1/9