Question

Given `(sqrt(x+1) - sqrt(2x+1)) / (sqrt(3x+4) - sqrt(2x+4) ]` how do you find the limit as x approaches 0?

Answer 1

Change the way it is written to avoid `0/0`.

Explanation:

By the time this problem is assigned, I assume students have seen things like

`lim_(xrarr0)(sqrt(9+x)-3)/x` which is found by 'rationalizing' the numerator:

`lim_(xrarr0)((sqrt(9+x)-3))/x*((sqrt(9+x)+3))/((sqrt(9+x)+3)) = lim_(xrarr0)1/((sqrt(9+x)-3)) = 1/6`

In this problem, we have to try something, so late's use the same trick on both the numerator and denominator

`(sqrt(x+1)-sqrt(2x+1))/(sqrt(3x+4)-sqrt(2x+4)) = ((sqrt(x+1)-sqrt(2x+1)))/((sqrt(3x+4)-sqrt(2x+4))) * ((sqrt(x+1)+sqrt(2x+1))(sqrt(3x+4)+sqrt(2x+4)))/((sqrt(x+1)+sqrt(2x+1))(sqrt(3x+4)+sqrt(2x+4))`

`= ((x+1-(2x+1))(sqrt(3x+4)+sqrt(2x+4)))/((x+4-4)(sqrt(x+1)+sqrt(2x+1)))`

`= (-x(sqrt(3x+4)+sqrt(2x+4)))/(x (sqrt(x+1)+sqrt(2x+1)))`

`= (-(sqrt(3x+4)+sqrt(2x+4)))/ (sqrt(x+1)+sqrt(2x+1))` `" "` (for `x != 0`)

`lim_(xrarr0)(sqrt(x+1)-sqrt(2x+1))/(sqrt(3x+4)-sqrt(2x+4)) = lim_(xrarr0)(-(sqrt(3x+4)+sqrt(2x+4)))/ (sqrt(x+1)+sqrt(2x+1))`

`= -(sqrt4+sqrt4)/(sqrt1+sqrt1) = (-4)/2=-2`

The same algebra with simplified notation gets us

`(sqrta - sqrtb)/(sqrtc - sqrtd) =((sqrta - sqrtb))/((sqrtc - sqrtd)) * ((sqrta + sqrtb)(sqrtc + sqrtd))/((sqrta + sqrtb)(sqrtc + sqrtd))`

`= ((a-b)(sqrtc+sqrtd))/((c-d)(sqrta+sqrtb))`