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The problem involves finding the limit:

\[
\lim_{x \to 1} \frac{(x - 4)(x - 1)}{(x - 1)(x + 3)}
\]

### Step 1: Simplifying the Expression

The expression can be simplified by canceling the common factor of \((x - 1)\) from the numerator and the denominator, as long as \(x \neq 1\) to avoid division by zero:

\[
\frac{(x - 4)(x - 1)}{(x - 1)(x + 3)} = \frac{x - 4}{x + 3} \quad \text{(for } x \neq 1)
\]

### Step 2: Substituting \(x = 1\)

Now, we can safely substitute \(x = 1\) into the simplified expression:

\[
\frac{1 - 4}{1 + 3} = \frac{-3}{4}
\]

Thus, the value of the limit is:

\[
\boxed{-\frac{3}{4}}
\]

### Conclusion

The correct answer is option **d: \(-\frac{3}{4}\)**.

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Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:
1. What is the importance of simplifying expressions before substituting in limits?
2. How do we handle indeterminate forms in limits, such as \( \frac{0}{0} \)?
3. Can we always cancel terms in the numerator and denominator in a limit?
4. How would you approach a limit problem if the result were an indeterminate form after simplification?
5. What other methods can be used to evaluate limits, such as L'Hopital's Rule?

**Tip**: Always check for common factors that can be canceled before substituting in a value when solving limits to avoid indeterminate forms.

lim_{x \to 1} \frac{(x - 4)(x - 1)}{(x - 1)(x + 3)}

This guide walks you through solving the limit lim_{x -> 1} (x-4)(x-1)/(x-1)(x+3), including simplifying expressions, canceling factors, and substituting values to find the final result of -3/4. Ideal for students in Grades 10-12 studying algebra and limits.

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