Final answer:
The limit of the function f(x) = ln(x)/x^99 as x approaches 0 is 0. This can be found by applying L'Hospital's Rule multiple times to reduce the power of x in the denominator.
Step-by-step explanation:
The given question involves finding the limit of the function f(x) = ln(x)/x^99 as x approaches 0. To evaluate this limit, we can use L'Hospital's Rule. Differentiating the numerator and denominator, we get f'(x) = (1/x) / (99x^98). Taking the limit as x approaches 0, we have:
lim f(x) = lim f'(x)
Substituting x=0 into the expression, we get lim f(x) = 0/0. This is an indeterminate form, so we can apply L'Hospital's Rule again. Differentiating the numerator and denominator once more, we have: f''(x) = (-1/x^2) / (99(98)x^97). Taking the limit as x approaches 0, we get:
lim f''(x) = 0/0
Again, we get an indeterminate form, so we can continue applying L'Hospital's Rule until we reach a limit that can be directly evaluated. Each application of L'Hospital's Rule will reduce the power of x in the denominator by 1. Since the denominator has a power of x^97 after two applications of L'Hospital's Rule, we need to apply it 97 more times. After applying L'Hospital's Rule 99 times, we will reach a limit of 0. Therefore, lim f(x) = 0 as x approaches 0.