Final answer:
To find the values of c such that the limit lim (x^3 - 1)/(x^c - 1) exists, we need to determine when the denominator becomes zero. The values of c that make the limit exist are c = 0 and c = 2πi, where i is an integer.
Step-by-step explanation:
To find the values of c such that the limit lim (x^3 - 1)/(x^c - 1) exists, we need to determine when the denominator becomes zero. In this case, the denominator becomes zero when x^c - 1 = 0. Solving for c, we have:
x^c = 1
Now, the exponent c is the power to which x must be raised to give 1. This only happens when c = 0 or when c is a multiple of 2πi, where i is an integer.
Therefore, the values of c that make the limit exist are c = 0 and c = 2πi, where i is an integer.