`lim_(x ->0) (sqrt(1 + 2x) - sqrt(1 - 4x))/x` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule...

`lim_(x ->0) (sqrt(1 + 2x) - sqrt(1 - 4x))/x` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`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