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
Step 1: Understand the condition for continuity. For a function f(x) to be continuous at a point x=c, the limit of the function as x approaches c must be equal to the function's value at c. That is, _x c f(x) = f(c). In this problem, the function is f(x) = (x)/( x) for x ≠ 0, and we need to redefine it at x=0 to make it continuous. So, we need to find the value of _x 0 f(x). Step 2: Evaluate the limit of the function as x approaches 0. The function is f(x) = (x)/( x). We need to find: _x 0 (x)/( x) This is a standard trigonometric limit. We know the fundamental limit: _x 0 ( x)/(x) = 1 We can rewrite the given limit using this identity: _x 0 (x)/( x) = _x 0 (1)/( x)x As x 0, the denominator ( x)/(x) approaches 1. Therefore: _x 0 (1)/( x)x = (1)/(1) = 1 Step 3: Determine the value for f(0) for continuity. For the function f(x) to be continuous at x=0, we must define f(0) to be equal to the limit we just calculated. Therefore, to make the function continuous at x=0, we must define f(0) = 1. The value that the function can be redefined to at x=0 to make it continuous is 1.