What is the value of the limit as x approaches 0 of (sin(x) - x)/x^3?
Correct: A
To evaluate the limit of (sin(x) - x)/x^3 as x approaches 0, we can use L'Hopital's Rule. L'Hopital's Rule states that if a limit is of the form 0/0, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting quotient. Applying L'Hopital's Rule once, we get lim (x→0) (cos(x) - 1)/3x^2. This limit is still of the form 0/0, so we apply L'Hopital's Rule again to get lim (x→0) (-sin(x))/6x. This limit is still of the form 0/0, so we apply L'Hopital's Rule again to get lim (x→0) (-cos(x))/6. Evaluating this limit, we get -1/6.