Evaluate `lim_(x rarr 0) [(x)/(1 - sqrt(1-x))]`.

1.	Evaluate `lim_(x rarr 0) [(x)/(1 - sqrt(1-x))]`.
Answer» When x = 0, the given function assumes the form `(0)/(1- sqrt(1-0))=(0)/(0)`, which is an indeterminate form. Since g(x), the denominator is an irrational form, we multiply the numerator and denominator with the rationalizing factor of g(x). `underset(x rarr 0)("lim")[(x)/(1 - sqrt(1-x))]=underset(x rarr 0)("lim")(x)/(1 - sqrt(1-x)) xx ((1 + sqrt(1-x)))/((1 + sqrt(1-x)))` `" "= underset(x rarr 0)("lim")(x(1 + sqrt(1-x)))/(1 - 1 + x)` `" "= underset(x rarr 0)("lim")(x(1 + sqrt(1-x)))/(x)` `" "= underset(x rarr 0)("lim")(1 + sqrt(1-x))=1 + sqrt(1) = 2`.

Discussion

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