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

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))#