Final Answer:
The limit as x approaches 3 for the function (x - 27)/(x^(1/3) - 3) is indeterminate, but it can be evaluated using L'Hôpital's Rule. After applying the rule, the limit simplifies to 1/9.
Step-by-step explanation:
To find the limit as x approaches 3 for the given function, we first substitute x = 3, resulting in (3 - 27)/(3^(1/3) - 3).
This expression yields 0/0, an indeterminate form. Applying L'Hôpital's Rule involves taking the derivative of the numerator and denominator until the indeterminate form changes. After one application, we get 1/(1/3 * 3^(-2/3)). Simplifying this expression, we obtain 1/9 as the final limit.
It's essential to note that L'Hôpital's Rule is applicable when the limit results in an indeterminate form, and the derivatives exist. In this case, the rule allows us to handle the indeterminate form and evaluate the limit accurately. The final result, 1/9, is the value approached by the function as x gets arbitrarily close to 3.