find the limit as x approaches 0 of (f(x+4) - f(4))/x if f(4) = 7

Question

the line y=3x-5 is tangent to the graph of y=f(x)  at the point (4,7)

Answer 1

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.