Question

For which of the following does lim f(x) = 0 ? x to infinity I. f(x) = lnx / x^99 II. f(x) = e^x / lnx III. f(x) = x^99 / e^x A. I only B. II only C. III only D. I and II only E. I and III only

Answer 1

To find the functions that satisfy the limit as x approaches infinity being equal to zero, we analyze each function individually. Functions I and III satisfy the given condition, resulting in answer option E: I and III only.

Step-by-step explanation:

To determine for which of the given functions the limit lim f(x) = 0 as x approaches infinity holds, we can use the concept of asymptotes. Let's analyze each function:

I. f(x) = lnx / x^99: Since the denominator, x^99, grows much faster than the numerator, lnx, as x approaches infinity, the overall function approaches zero. Therefore, option I satisfies the given condition.

II. f(x) = e^x / lnx: As x approaches infinity, the exponential function e^x grows much faster than the logarithmic function lnx. Hence, the ratio f(x) approaches infinity and does not satisfy the given condition.

III. f(x) = x^99 / e^x: In this case, the numerator, x^99, grows at a slower rate compared to the denominator, e^x, as x approaches infinity. Thus, the overall function approaches zero and satisfies the given condition.

Based on these analyses, the functions that satisfy the condition lim f(x) = 0 as x approaches infinity are: I and III. Therefore, the correct answer is E. I and III only.

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