Final answer:
To find the constant c such that the limit of the function [(x² + 3x + c) / (x² - 15x + 56)] exists as x approaches 8, factor the numerator and denominator to simplify the function. Set the factor of (x - 8) equal to zero in the numerator to find the value of c. The limit of the function with c = -8 is 1/6.
Step-by-step explanation:
To find the constant c such that the limit of the function [(x² + 3x + c) / (x² - 15x + 56)] exists as x approaches 8, we can simplify the function by factoring the numerator and denominator. The factored function becomes [(x + 8)(x + c)] / [(x - 7)(x - 8)]. In order for the limit to exist, there should be no factor of (x - 8) in the numerator. Therefore, c should be equal to -8. The limit of the function as x approaches 8 with c = -8 is 1/6.