The limit of (1-cos(2x))/(5x) as x approaches 0 is 0, which is found using L'Hôpital's Rule.
The student asks about the limit as x approaches 0 for the expression (1-cos(2x))/(5x). This is a common type of limit problem in calculus which involves evaluating an indeterminate form 0/0. To solve it, we can apply L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value c results in an indeterminate form, then it is equal to the limit of f'(x)/g'(x) under certain conditions.
First, we find the derivatives of the numerator and denominator separately. The derivative of 1-cos(2x) with respect to x is 2sin(2x), and the derivative of 5x with respect to x is 5. Applying L'Hôpital's Rule, the limit becomes the limit of (2sin(2x))/5 as x approaches 0. The sine function is continuous, and sin(0) is 0, so the limit simplifies directly to 0.
So, the answer is A. 0, which is the result you obtain after applying L'Hôpital's Rule and substituting x=0 into the derivative.