l = lim as x approaches 0 of x/(the square root of (1+x) - the square root of (1-x)
decide whether:
l=-1
or
l=0
or
l=1
Let me make sure I understand the question. Do we have
lim x->0 x/[sqrt(1+x) - sqrt(1-x)] ?
If so then multiply the expression by
[sqrt(1+x) + sqrt(1-x)]/[sqrt(1+x) + sqrt(1-x)]
to get
(x*[sqrt(1+x) + sqrt(1-x)])/[(1+x) - (1-x)]=
(x*[sqrt(1+x) + sqrt(1-x)])/2x=
=limx->0 [sqrt(1+x) + sqrt(1-x)]/2=
something you can do.
biz
Yes, you have understood the question correctly. To determine the value of the limit, we will simplify the expression by multiplying it by the conjugate of the denominator.
The conjugate of the denominator is [sqrt(1+x) + sqrt(1-x)], so we multiply the expression by this conjugate on both the numerator and denominator:
lim x->0 x/[sqrt(1+x) - sqrt(1-x)] * [sqrt(1+x) + sqrt(1-x)]/[sqrt(1+x) + sqrt(1-x)]
When we multiply the numerator, we get x * [sqrt(1+x) + sqrt(1-x)]. In the denominator, the terms (1+x) and (1-x) cancel out, leaving us with 2x:
[x * (sqrt(1+x) + sqrt(1-x))] / 2x
Then, we simplify the expression further by canceling out the common factor of x:
(sqrt(1+x) + sqrt(1-x)) / 2
Finally, we can take the limit of this simplified expression as x approaches 0:
lim x->0 (sqrt(1+0) + sqrt(1-0)) / 2
= (sqrt(1) + sqrt(1)) / 2
Since sqrt(1) is equal to 1, we have:
(1 + 1) / 2
= 2 / 2
= 1
Therefore, the value of the limit is l=1.