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: $-\frac{3}{4}$.

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.