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Question
Evaluate the following:
`lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x`
Solution
`lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x`
= `lim_(x->0) (sqrt(1+x) - sqrt(1-x))/(5x) xx ((sqrt(1+x) + sqrt(1-x))/(sqrt(1+x) + sqrt(1-x)))`
= `lim_(x->0) ((1+x)-(1-x))/(5x(sqrt(1+x) + sqrt(1-x)))`
= `lim_(x->0) ((1+x-1+x))/(5x(sqrt(1+x) + sqrt(1-x)))`
= `lim_(x->0) (2x)/(5x(sqrt(1+x) + sqrt(1-x)))`
= `lim_(x->0) (2)/(5(sqrt(1+0) + sqrt(1-0)))`
`= 2/(5(1+1)) = 1/5`
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