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: Analyze function (A) f(x) = (1)/(|x|). The function is not defined at x=0 because |0| is undefined. Let's evaluate the limit as x 0: As x 0, |x| 0^+. As y 0^+, y -. So, _x 0 |x| = -. Therefore, _x 0 f(x) = _x 0 (1)/(|x|) = (1)/(-) = 0. Since the limit exists but the function is not defined at x=0, this is a removable discontinuity. Step 2: Analyze function (B) f(x) = ((| x|)/(x)). The function is not defined at x=0 because the denominator x is zero. Let's evaluate the limit as x 0: Consider the inner function g(x) = (| x|)/(x). For x > 0 (as x 0^+), x > 0, so | x| = x. _x 0^+ ( x)/(x) = 1. For x < 0 (as x 0^-), x < 0, so | x| = - x. _x 0^- (- x)/(x) = -1. Now, evaluate the limit of f(x): _x 0^+ f(x) = _x 0^+ ((| x|)/(x)) = (1). _x 0^- f(x) = _x 0^- ((| x|)/(x)) = (-1) = (1). Since the left-hand limit and right-hand limit are equal, _x 0 f(x) = (1). Since the limit exists but the function is not defined at x=0, this is a removable discontinuity. Step 3: Analyze function (C) f(x) = x (()/(x)). The function is not defined at x=0 because ()/(x) is undefined. Let's evaluate the limit as x 0: We know that -1 (()/(x)) 1. Multiplying by x: If x > 0, then -x x (()/(x)) x. If x < 0, then x x (()/(x)) -x. In both cases, |x (()/(x))| |x|. As x 0, |x| 0. By the Squeeze Theorem, _x 0 x (()/(x)) = 0. Since the limit exists but the function is not defined at x=0, this is a removable discontinuity. Step 4: Analyze function (D) f(x) = (1)/(1+2^ x). The function is not defined at x=0 because x = ( x)/( x) is undefined at x=0. Let's evaluate the limit as x 0: Consider the behavior of x as x 0: As x 0^+, x 0^+ and x 1, so x +. As x 0^-, x 0^- and x 1, so x -. Now, evaluate the limits of f(x): _x 0^+ f(x) = _x 0^+ (1)/(1+2^ x). As x +, 2^ x +. So, _x 0^+ f(x) = (1)/(1+) = 0. _x 0^- f(x) = _x 0^- (1)/(1+2^ x). As x -, 2^ x 0. So, _x 0^- f(x) = (1)/(1+0) = 1. Since the left-hand limit (1) and the right-hand limit (0) are different, the limit _x 0 f(x) does not exist. This indicates an irremovable (jump) discontinuity. Step 5: Conclude based on the analysis. All four functions are not defined at x=0. Functions (A), (B), and (C) have removable discontinuities at x=0 because their limits exist. Function (D) has an irremovable (jump) discontinuity at x=0 because its left-hand and right-hand limits are different. The question asks for the function(s) not defined at x=0 that has/have an irremovable discontinuity at x=0. Only function (D) fits this description. The final answer is D. That's 2 down. 3 left today — send the next one.