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The given function is:

\[
f(x) = \frac{1 - x}{(x + 1)(x + 3)}
\]

You are asked to compute:

\[
\lim_{x \to -1} f(x)
\]

Let's first substitute \(x = -1\) directly into the function to see if the limit can be evaluated directly or if it leads to an indeterminate form:

\[
f(-1) = \frac{1 - (-1)}{(-1 + 1)((-1) + 3)} = \frac{1 + 1}{(0)(2)} = \frac{2}{0}
\]

Since the denominator is zero and the numerator is not zero, this suggests a **vertical asymptote** at \(x = -1\). Therefore, the limit likely approaches either positive or negative infinity, depending on the sign of the function around \(x = -1\).

### Behavior around \(x = -1\):

- For \(x \to -1^-\) (approaching from the left):
  - The factor \(x + 1\) is **negative**.
  - The factor \(x + 3\) is **positive**.

Thus, the denominator will be negative, and since the numerator is positive (\(1 - x > 0\)), the function will approach **negative infinity** as \(x \to -1^-\).

- For \(x \to -1^+\) (approaching from the right):
  - The factor \(x + 1\) is **positive**.
  - The factor \(x + 3\) is **positive**.

So the denominator will be positive, and the numerator remains positive, leading the function to approach **positive infinity** as \(x \to -1^+\).

### Conclusion:

Since the limit does not approach the same value from both sides, the limit **does not exist**.

Would you like any further details or have any additional questions?

Here are some related questions for practice:
1. How do you determine if a function has a vertical asymptote?
2. What is the significance of one-sided limits in determining the overall limit?
3. How does the behavior of the numerator and denominator affect the sign of the limit?
4. What is the formal definition of a limit not existing?
5. Can you calculate limits using L'Hopital's Rule in cases of indeterminate forms?

**Tip:** Always check both one-sided limits when the denominator approaches zero to determine if the limit exists or results in infinity.

For the function f(x) = (1 - x) / ((x + 1)(x + 3)), compute the limit lim x → -1 f(x).

This problem explores the limit of a rational function where x approaches -1. Learn how to determine vertical asymptotes and apply one-sided limits to conclude whether the limit exists. This is suitable for students in grades 11-12 or introductory college-level calculus.

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