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When dealing with polynomial expressions, factoring is an essential skill. Factoring a polynomial means expressing it as a product of simpler polynomials. This is similar to expressing a number as a product of its prime factors. Polynomial expressions often have more than one term, so we look for ways to group these terms and then factor them.
For instance, in our problem, the polynomial in the numerator is \(x^{2} - 5x + 4\). This polynomial is a quadratic, commonly written as \(ax^2 + bx + c\). To factor it, we need to find two numbers that multiply to 4 (the constant term) and add to -5 (the linear coefficient). These numbers are -4 and -1, leading to the factors \((x - 4)(x - 1)\).
Similarly, the denominator \(x^{2} - 2x - 8\) can be factored into \((x - 4)(x + 2)\) by finding two numbers that multiply to -8 and add to -2, which are -4 and 2.

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions involves reducing them to their lowest terms. This is done by canceling out any common factors between the numerator and the denominator.
In the exercise, after factoring, we have the expression:\[\frac{(x - 4)(x - 1)}{(x - 4)(x + 2)}\]
Here, \((x - 4)\) is a common factor in both the numerator and the denominator. Canceling these terms simplifies the rational expression to \(\frac{x - 1}{x + 2}\).
It's crucial to perform this step because it often eliminates terms that cause undefined behavior in the function, such as division by zero, allowing for proper evaluation of limits.

Once we have simplified the rational expression, we can employ the substitution method to find limits. The substitution method involves simply putting the value of \(x\) into the simplified expression when the limit is evaluated.
In this specific example, after simplifying, we substitute \(x = 4\) into the expression \(\frac{x - 1}{x + 2}\).
Calculating, we get:

• \(x - 1 = 4 - 1 = 3\)
• \(x + 2 = 4 + 2 = 6\)

This simplifies to \(\frac{3}{6} = 0.5\).
The substitution method provides an easy way to compute the limit once the problematic factors causing undefined behavior are canceled out. It is a straightforward technique when the rational expression is continuous and defined at the point of interest.