Question

How do you simplify `(sqrt(4+h)-2)/h` ?

Answer 1

`(sqrt(4+h)-2)/h = 1/(sqrt(4+h)+2)`

with exclusion `h != 0`

Explanation:

Given:

`(sqrt(4+h)-2)/h`

We find:

`(sqrt(4+h)-2)/h = ((sqrt(4+h)-2)(sqrt(4+h)+2))/(h(sqrt(4+h)+2))`

`color(white)((sqrt(4+h)-2)/h) = ((4+h)-4)/(h(sqrt(4+h)+2))`

`color(white)((sqrt(4+h)-2)/h) = color(red)(cancel(color(black)(h)))/(color(red)(cancel(color(black)(h)))(sqrt(4+h)+2))`

`color(white)((sqrt(4+h)-2)/h) = 1/(sqrt(4+h)+2)`

`color(white)()`
Footnote

This is the sort of expression you find involved in a limit problem, like:

What is: `lim_(h->0) (sqrt(4+h)-2)/h` ?

The tricky thing here is that both the numerator and denominator become `0` when `h = 0`, but we can find the limit using our simplification...

`lim_(h->0) (sqrt(4+h)-2)/h = lim_(h->0) 1/(sqrt(4+h)+2)`

`color(white)(lim_(h->0) (sqrt(4+h)-2)/h) = 1/(sqrt(4+0)+2)`

`color(white)(lim_(h->0) (sqrt(4+h)-2)/h) = 1/(2+2)`

`color(white)(lim_(h->0) (sqrt(4+h)-2)/h) = 1/4`