Limit definition

Question

For p(z)=1/z+2 use the limit definition of the derivative to find the derivatives below.

p'(-6)

p'(-3)

p'(-1)

p'(2)

Answer 1

I'll solve the first one, then you can use the pattern to do the rest.

I'm using the second limit derivative definition.

f'(x₀) =lim x→ x₀ (f(x) - f(x₀))/(x-x₀)

f'(-6) = lim x→ -6 (f(x) - f(-6))/(x-(-6)) = lim x→ -6 (1/x+2 - (1/-6 +2))/(x+6)

= lim x→ -6 (1/x +2 - 1/-6 -2)/(x+6) = lim x→ -6 (1/x+6)/(x+6) = lim x→ -6 (x+6)/(6x(x+6))

= lim x→ -6 1/6x = 1/(6(-6)) = -1/36