To get from the first line to the second, we multiplied the two fractions by multiplying numerators and denominators, and we used the identity
To get from the second line to the third, we used the fact that the squaring function is the inverse of the square root function. To get from the third line to the fourth, we simplified the numerator to `Delta x` and canceled this `Delta x` with the one in the denominator.
As `Delta x -> 0`, `sqrt(x+Delta x)+sqrt(x)` approaches the limiting value `2sqrt(x)`. Thus `1/(sqrt(x+Delta x)+sqrt(x)` approaches the limiting value `1/(2sqrt(x))`. This shows that