This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To evaluate the limit, we first need to analyze the domain of the function and the behavior of its argument as x approaches 0. Step 1: Determine the domain of the function. The natural logarithm (u) is defined only when u > 0. In this case, u = (1)/(x^2 - 5x). So, we must have (1)/(x^2 - 5x) > 0. This implies that x^2 - 5x must be positive. Factoring the denominator, we get x(x-5) > 0. This inequality holds when both factors are positive (x > 0 and x-5 > 0 x > 5) or when both factors are negative (x < 0 and x-5 < 0 x < 0). Thus, the domain of the function is (-, 0) (5, ). Step 2: Evaluate the left-hand limit as x 0^-. As x approaches 0 from the left side (x 0^-), x is a small negative number. For example, if x = -0.001, then x-5 = -5.001. So, x(x-5) is (negative) × (negative) = positive. Therefore, as x 0^-, x^2 - 5x 0^+. This means (1)/(x^2 - 5x) (1)/(0^+) = +. Then, the left-hand limit is: _x 0^- ((1)/(x^2 - 5x)) = _y (y) = Step 3: Evaluate the right-hand limit as x 0^+. As x approaches 0 from the right side (x 0^+), x is a small positive number. For example, if x = 0.001, then x-5 = -4.999. So, x(x-5) is (positive) × (negative) = negative. Therefore, as x 0^+, x^2 - 5x 0^-. This means (1)/(x^2 - 5x) (1)/(0^-) = -. Since the natural logarithm is not defined for negative arguments, (-) is undefined. Thus, the right-hand limit _x 0^+ ((1)/(x^2 - 5x)) does not exist. Step 4: Conclude the overall limit. For the overall limit _x 0 f(x) to exist, both the left-hand limit and the right-hand limit must exist and be equal. Since the right-hand limit does not exist (the function is undefined for values of x approaching 0 from the right), the overall limit does not exist. The final answer is Does not exist. Send me the next one 📸