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

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

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

`=lim_(x->0)(6x)/(x(sqrt(1+2x)+sqrt(1-4x)))`

`=lim_(x->0)6/(sqrt(1+2x)+sqrt(1-4x))`

plug in the value,

`=6/(sqrt(1)+sqrt(1))=6/2`

=**3**

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