Lim_x->0 (sqrt(1+x2) - sqrt(1-x)) / x2

Step 1: Check for indeterminate form by substituting x=0 into the expression. Numerator: sqrt(1+0^2) - sqrt(1-0) = sqrt(1) - sqrt(1) = 1-1 = 0. Denominator: 0^2 = 0. Since the result is (0)/(0), it is an indeterminate form, and further steps are needed. Step 2: Multiply the numerator and denominator by the conjugate of the numerator. The conjugate of sqrt(1+x^2) - sqrt(1-x) is sqrt(1+x^2) + sqrt(1-x). _x 0 ( sqrt(1+x^2) - sqrt(1-x)x^2 × sqrt(1+x^2) + sqrt(1-x)sqrt(1+x^2) + sqrt(1-x) ) Step 3: Simplify the numerator using the difference of squares formula, (a-b)(a+b) = a^2 - b^2. _x 0 (sqrt(1+x^2))^2 - (sqrt(1-x))^2x^2(sqrt(1+x^2) + sqrt(1-x)) _x 0 ((1+x^2) - (1-x))/(x^2(sqrt(1+x^2) + 1-x)) Step 4: Expand and simplify the numerator. _x 0 (1+x^2 - 1 + x)/(x^2(sqrt(1+x^2) + 1-x)) _x 0 (x^2 + x)/(x^2(sqrt(1+x^2) + 1-x)) Step 5: Factor out x from the numerator and cancel a common factor of x with the denominator. _x 0 (x(x+1))/(x^2(sqrt(1+x^2) + 1-x)) _x 0 (x+1)/(x(sqrt(1+x^2) + 1-x)) Step 6: Evaluate the limit by substituting x=0 into the simplified expression. Numerator: 0+1 = 1. Denominator: 0 × (sqrt(1+0^2) + sqrt(1-0)) = 0 × (sqrt(1) + sqrt(1)) = 0 × (1+1) = 0 × 2 = 0. The expression is of the form (1)/(0), which indicates that the limit is either , -, or does not exist. Step 7: Analyze the one-sided limits. The term (sqrt(1+x^2) + sqrt(1-x)) approaches 2 as x 0, which is a positive value. The sign of the denominator is determined by the sign of x. • As x 0^+ (x approaches 0 from the positive side), x > 0. The denominator x(sqrt(1+x^2) + sqrt(1-x)) approaches 0 × 2 = 0 from the positive side (0^+). So, _x 0^+ (x+1)/(x(sqrt(1+x^2) + 1-x)) = (1)/(0^+) = +. • As x 0^- (x approaches 0 from the negative side), x < 0. The denominator x(sqrt(1+x^2) + sqrt(1-x)) approaches 0 × 2 = 0 from the negative side (0^-). So, _x 0^- (x+1)/(x(sqrt(1+x^2) + 1-x)) = (1)/(0^-) = -. Since the left-hand limit (-) and the right-hand limit (+) are not equal, the limit does not exist. The final answer is The limit does not exist. That's 2 down. 3 left today — send the next one.