Evaluate `lim_(xto0) (sqrt(2+x)-sqrt(2))/(x).`

1.	Evaluate `lim_(xto0) (sqrt(2+x)-sqrt(2))/(x).`
Answer» When `xto0,` the expression `underset(xto0)lim(sqrt(2+x)-sqrt(2))/(x).` takes the form 0/0. Rationalizing the numerator, we have `underset(xto0)lim(sqrt(2+x)-sqrt(2))/(x)=underset(xto0)lim((sqrt(2+x)-sqrt(2))(sqrt(2+x)+sqrt(2)))/(x(sqrt(2+x)+sqrt(2)))` `=underset(xto0)lim(2+x-2)/(x(sqrt(2+x)+sqrt2))` `=underset(xto0)lim(1)/(sqrt(2+x)+sqrt(2))=(1)/(2sqrt(2))`

Discussion

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