Replace x with δx then (f(4+δx)-f(4))/δx=δf/δx.
But as δx→0, δf/δx→df/dx.
The line y=3x-5 passes through (4,7) and so does f(x), and the slope of the tangent line is df/dx at that point, so df/dx=3.
This must therefore be the limit as δx→0, or, as x→0 in the original expression.
So (f(4+x)-f(4))/x=3 as x→0.