Question

What is the limit x-->0 (sqrt(1+x)-sqrt(1-x))/x ?

Accepted Answer

You should substitute 0 for x such that:
lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x = (sqrt(1+0) - sqrt(1-0))/0
lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x = (1-1)/0 = 0/0
The indetermination 0/0 could be solved using l'Hospital's theorem such that:
lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x = lim_(x->0) ((sqrt(1+x) - sqrt(1-x))')/(x') 
lim_(x->0) ((sqrt(1+x) - sqrt(1-x))')/(x') = lim_(x->0) 1/(2sqrt(1+x))+ 1/(2sqrt(1-x))
Substituting 0 for x yields:
lim_(x->0) 1/(2sqrt(1+x)) + 1/(2sqrt(1-x)) = 1/(2sqrt(1+0)) + 1/(2sqrt(1-0)) 
lim_(x->0) 1/(2sqrt(1+x)) + 1/(2sqrt(1-x)) = 1/2 + 1/2 = 1
Hence, evaluating the given limit under the given conditions yields lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x = 1 .

Answer

The limit limit lim_(x->0) (sqrt(1+x)-sqrt(1-x))/x has to be determined.
Substituting x = 0 in (sqrt(1+x)-sqrt(1-x))/x gives the indeterminate result 0/0 .
Let us alter the expression by multiplying it with its conjugate.
lim_(x->0) (sqrt(1+x)-sqrt(1-x))/x
= lim_(x->0) ((sqrt(1+x)-sqrt(1-x))*(sqrt(1+x)+sqrt(1-x)))/(x*(sqrt(1+x)+sqrt(1-x))) 
Use the relation (x - a)(x + a) = x^2 - a^2
= lim_(x->0) ((sqrt(1+x))^2-(sqrt(1-x))^2)/(x*(sqrt(1+x)+sqrt(1-x)))
= lim_(x->0) (1+x - 1 + x)/(x*(sqrt(1+x)+sqrt(1-x)))
= lim_(x->0) (2x)/(x*(sqrt(1+x)+sqrt(1-x)))
= lim_(x->0) (2)/((sqrt(1+x)+sqrt(1-x)))
Now when we substitute x = 0 the result is (2)/(sqrt 1 + sqrt 1) = 2/(2*sqrt 1) = 1
The limit lim_(x->0) (sqrt(1+x)-sqrt(1-x))/x = 1

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What is the limit x-->0 (sqrt(1+x)-sqrt(1-x))/x ?

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