Question

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

Answer 1

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Explanation:

If you want `lim_(xrarr0)(sqrt(1+2x)-sqrt(1-4x)/x)`

`lim_(xrarr0)sqrt(1+2x) = 1` and

`lim_(xrarr0)sqrt(1-4x)/x` does not exist, because as `xrarr0^+`, we have `sqrt(1-4x)/x` increases without bound.

Therefore, the difference, `lim_(xrarr0)(sqrt(1+2x)-sqrt(1-4x)/x)` does not exist because as `xrarr0^+`, the value of the expression decreases without bound.

If you want `lim_(xrarr0)(sqrt(1+2x)-sqrt(1-4x))/x` then "rationalize" the numerator.

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

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

`= (6x)/(x(sqrt(1+2x)+sqrt(1-4x)))`

`= 6/(sqrt(1+2x)+sqrt(1-4x))`

Now we have

`lim_(xrarr0)(sqrt(1+2x)-sqrt(1-4x))/x = lim_(xrarr0)6/(sqrt(1+2x)+sqrt(1-4x))`

`> 6/(sqrt1+sqrt1) = 3`