https://file.aiquickdraw.com/mathbot/file/1726699185739r7c87qzj.jpg

Let's break down and solve the given problem step by step. The function provided is:

\[
f(x) = \frac{x^2 - 3x}{x^3 - 3x}
\]

### (a) Determine \( \lim_{x \to 3} f(x) \). (Hint: Use factorization.)

We begin by simplifying the function to evaluate the limit.

1. **Factorize the numerator** \( x^2 - 3x \):
   \[
   x^2 - 3x = x(x - 3)
   \]

2. **Factorize the denominator** \( x^3 - 3x \):
   \[
   x^3 - 3x = x(x^2 - 3)
   \]
   which can be factored as:
   \[
   x(x - \sqrt{3})(x + \sqrt{3})
   \]
   So, the factored form of the function is:
   \[
   f(x) = \frac{x(x -

Given f(x) = (x^2 - 3x) / (x^3 - 3x): (a) Determine lim (x -> 3) f(x) (Hint: Use factorization). (b) Evaluate f(3). (c) Is f(x) continuous or discontinuous at x = 3? Support your answer using the condition for continuity at x = 3. (d) Write the intervals where f(x) is continuous.

This problem explores the limit, factorization, and continuity of the function f(x) = (x^2 - 3x) / (x^3 - 3x). Learn to find lim (x -> 3) f(x) using factorization, evaluate f(3), and determine the function's continuity at x = 3. Suitable for students learning advanced calculus.

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