We are calculating the slope of the function f(x) = sqrt(x) at x = 4.
f(4) = 2
f(4+h) = sqrt(4 + h)
[f(4+h) - f(4)]/h =?
You have to pick a small h,and calculate it. Then pick an even smaller value and calculate it again. That's called taking the limit as h-> 0
If you pick h=0.1, you get a slope of
If you pick h=0.01, you get a slope of 0.2498
Try an even lower value of h and you will see the trend to the limit.
The exact value of the slope, using the formula for the derivative at x=4, is (1/2)/(sqrt 4) = 1/4