Evaluate
lim as x-> 0
square root (x+1) - square root (2x+1)
/ square root (3x+4) - square root (2x+4)
To evaluate the limit as x approaches 0 for the given expression, we can use a common technique called rationalization. The goal is to eliminate any radicals from the denominator.
Let's start by rationalizing the denominator. Multiply both the numerator and the denominator by the conjugate of the denominator, which is the conjugate of square root (3x+4) - square root (2x+4), i.e., square root (3x+4) + square root (2x+4).
This yields:
[(square root (x+1) - square root (2x+1)) * (square root (3x+4) + square root (2x+4))]
/ [(square root (3x+4) - square root (2x+4)) * (square root (3x+4) + square root (2x+4))]
Now, apply the difference of squares formula to the denominator:
[(square root (x+1) - square root (2x+1)) * (square root (3x+4) + square root (2x+4))]
/ [(3x+4) - (2x+4)]
Simplify:
[(square root (x+1) - square root (2x+1)) * (square root (3x+4) + square root (2x+4))]
/ (x)
Next, factor out a common term from the numerator:
[square root (x+1) - square root (2x+1)] * [square root (3x+4) + square root (2x+4)] / (x)
Now, apply the limit as x approaches 0.
When we substitute x = 0 into the expression, we get:
[square root (0+1) - square root (2*0+1)] * [square root (3*0+4) + square root (2*0+4)] / (0)
[square root (1) - square root (1)] * [square root (4) + square root (4)] / (0)
(0) * (2+2) / (0)
0 / 0
At this point, we have an indeterminate form, which means we cannot directly evaluate the limit. To further simplify, we can apply L'Hopital's rule by finding the derivative of both the numerator and the denominator with respect to x and then taking the limit again.