The limit definition is:

#lim_(hto0)(f(x + h) - f(x))/h#

We compute #f(x + h)#:

#f(x + h) = (x + h)/(x + h + 4)#

Substitute #f(x + h) and f(x)# into the definition:

#lim_(hto0)((x + h)/(x + h + 4) - (x)/(x + 4))/h#

Multiply the expression by 1 in the form of #((x + h + 4)(x + 4))/((x + h + 4)(x + 4))#

#lim_(hto0)((x + h)/(x + h + 4) - (x)/(x + 4))/h((x + h + 4)(x + 4))/((x + h + 4)(x + 4)) =#

#lim_(hto0)(((x + h)(x + h + 4)(x + 4))/(x + h + 4) - (x(x + h + 4)(x + 4))/(x + 4))/(h(x + h + 4)(x + 4)) =#

#lim_(hto0)((x + h)(x + 4) - (x(x + h + 4)))/(h(x + h + 4)(x + 4)) = #

#lim_(hto0)((x^2 + 4x + hx + 4h) - (x^2 + hx + 4x))/(h(x + h + 4)(x + 4)) = #

#lim_(hto0)(4h)/(h(x + h + 4)(x + 4)) = #

#lim_(hto0)4/((x + h + 4)(x + 4))#

Now it is ok to let h become 0:

#4/((x + 4)(x + 4)) =#

#4/(x + 4)^2#