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
A removable discontinuity occurs at a point x=c if the function F(c) is undefined, but the limit _x c F(x) exists. Let's analyze the given example: F(x) = (x^2 - 6x + 5)/(x^2 - 4x + 3) at c=1. Step 1: Evaluate F(c). Substitute x=1 into the function: F(1) = (1^2 - 6(1) + 5)/(1^2 - 4(1) + 3) = (1 - 6 + 5)/(1 - 4 + 3) = (0)/(0) Since we have an indeterminate form (0)/(0), F(1) is undefined. Step 2: Evaluate the limit as x c. To evaluate the limit _x 1 (x^2 - 6x + 5)/(x^2 - 4x + 3), we factor the numerator and the denominator. Factor the numerator x^2 - 6x + 5: We look for two numbers that multiply to 5 and add to -6. These are -1 and -5. So, x^2 - 6x + 5 = (x-1)(x-5). Factor the denominator x^2 - 4x + 3: We look for two numbers that multiply to 3 and add to -4. These are -1 and -3. So, x^2 - 4x + 3 = (x-1)(x-3). Now, substitute the factored forms into the limit expression: _x 1 ((x-1)(x-5))/((x-1)(x-3)) Since x 1, x ≠ 1, so we can cancel the (x-1) terms: _x 1 (x-5)/(x-3) Now, substitute x=1 into the simplified expression: (1-5)/(1-3) = (-4)/(-2) = 2 The limit _x 1 F(x) exists and is equal to 2. Step 3: Conclude the type of discontinuity. Since F(1) is undefined but _x 1 F(x) exists (it equals 2), the function F(x) has a removable discontinuity at x=1. The steps shown in your notes are correct. The function F(x) = (x^2 - 6x + 5)/(x^2 - 4x + 3) has a removable discontinuity at x=1. The value that would make the function continuous at x=1 is 2. Send me the next one 📸