Verified answer

Rationalize the numerator by multiplying the numerator and denominator by 1 + √(1 - x^2) to get:

lim (x-->0) [1 - √(1 - x^2)]/x

= lim (x-->0) {[1 + √(1 - x^2)][1 - √(1 - x^2)]}/{x[1 + √(1 - x^2)]}

= lim (x-->0) [1 - (1 - x^2)]/{x[1 + √(1 - x^2)]}, via difference of squares

= lim (x-->0) x^2/{x[1 + √(1 - x^2)]}, by simplifying the numerator

= lim (x-->0) x/[1 + √(1 - x^2)], by canceling the conflicting factor of x

= 0/(1 + 1), by evaluating the result at x = 0

= 0.

I hope this helps!

multiply both side by 1+√(1-x^2)

1-(1-x^2)/[x( 1+√(1-x^2))]

x^2/[(x)( 1+√(1-x^2))]

x/[1+√(1-x^2)]

let x=0

0/1

=0 is the limit

Divide everything by x.