Evaluate `lim_(xto0){(sqrt(1+x)-sqrt(1-x))/(x)}.`

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...Good day, An alternative way to solve the same indeterminate limit is as follows: lim(x-->0){(sqrt(1 + x) - sqrt(1 - x))/x} (multiply numerator and denominator by 2) = lim(x-->0){2(sqrt(1 + x) - sqrt(1 - x))/2x)} (rewrite denominator 2x as follows: 2x = (1 + x) - (1 - x)) = lim(x-->0){2(sqrt(1 + x) - sqrt(1 - x))/((1 + x) - (1 - x))} (treat denominator as a difference of squares: (1 + x) - (1 - x) = (sqrt(1 + x) - sqrt(1 - x))(sqrt(1 + x) + sqrt(1 - x)), and then cancel the factor (sqrt(1 + x) - sqrt(1 - x)) of numerator and denominator) = lim(x-->0){2/(sqrt(1 + x) + sqrt(1 - x))} (evaluate the solvable limit) = 2/(sqrt(1) + sqrt(1)) = 2/2 = 1... I hope you appreciate this method too, Thank you for your math efforts, and take care, Jan-W

jan-willemreens

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