Final answer:
To find the limit of (x)/(1-cos(x))2 as x approaches 0, we make use of L'Hôpital's Rule. After applying the rule twice, we find that the limit evaluates to 1/2.
Step-by-step explanation:
The student asked to find the limit as x approaches 0 of the expression (x)/(1-cos(x))2. Unfortunately, the information provided in the question does not directly help in finding this limit, and standard calculus techniques such as factoring, rationalization, or special trigonometric limits must be employed instead.
Using L'Hôpital's Rule, we can find this limit since the initial form is an indeterminate form 0/0. We take the derivative of both the numerator and the denominator and then find the limit. The derivative of x is 1, and the derivative of (1-cos(x))2 using the chain rule is 2(1-cos(x))(-sin(x)). Applying L'Hôpital's Rule gives us the new expression to evaluate as x approaches 0: limit x->0 of 1/(-2sin(x)(1-cos(x))).
We know that sin(x) approaches 0 and (1-cos(x)) approaches 0 as x approaches 0, so if we apply L'Hôpital's Rule a second time, we end up with the limit of 1/(2(cos(x)+sin^2(x)/cos(x))) as x approaches 0. Evaluated, this limit gives us 1/2, since cos(0) is 1 and sin(0) is 0. Therefore, the final limit is 1/2.