Final answer:
To find the limits of the function f(x) = x / ln(x), we evaluate the limits at different values of x, including x → 0⁺, x → 1⁻, and x → 1⁺. The limit at x → 0⁺ is infinite, the limit at x → 1⁻ is 0, and the limit at x → 1⁺ is infinite. Therefore, the correct answer is option C) The limits at x → 0⁺ and x → 1⁺ are finite, and the limit at x → 1⁻ is infinite.
Step-by-step explanation:
To find the limits of the function f(x) = x / ln(x), we will evaluate the limits at different values of x.
(i) To find the limit as x approaches 0 from the right (x → 0⁺), we substitute a small positive value for x in the function. As x gets closer to 0, the value of ln(x) approaches negative infinity, causing the function to approach positive infinity. So, the limit is infinite.
(ii) To find the limit as x approaches 1 from the left (x → 1⁻), we substitute values slightly less than 1 in the function. As x gets closer to 1, the value of ln(x) approaches 0, and the function approaches 0. So, the limit is 0.
(iii) To find the limit as x approaches 1 from the right (x → 1⁺), we substitute values slightly greater than 1 in the function. The value of ln(x) is positive, and as x approaches 1, the function approaches positive infinity. So, the limit is infinite.
Therefore, the correct answer is option C) The limits at x → 0⁺ and x → 1⁺ are finite, and the limit at x → 1⁻ is infinite.