`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.

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Chapter 4, 4.4 - Problem 23 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`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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