91Ó°ÊÓ

Rational functions are expressions that are formed by the ratio of two polynomials. For example, in the exercise, the rational function is defined as \( \frac{x^2 - 8}{x - 2} \). These functions are very common in calculus, and understanding them is essential for solving limits and other calculus problems.
What makes rational functions interesting is how they behave near certain values, especially where the denominator becomes zero, causing undefined behavior. However, as in this problem, evaluating the function at \( x = 3 \) does not cause this undefined behavior since the denominator remains non-zero.
Always keep an eye on potential cases where the denominator might be zero. This knowledge helps us anticipate any possible indeterminate forms and apply the correct strategies for solving limits.

Direct substitution is one of the simplest methods for finding limits of a function. When we talk about direct substitution, we mean plugging the value that \( x \) approaches directly into the function. This method only works if doing so does not lead to an indeterminate form.
In our exercise, after the analysis, we realized that substituting \( x = 3 \) directly into the function did not lead to any problematic situations, allowing us to smoothly calculate the limit:

• We simply replaced \( x \) with 3, which led to an easy computation: \( \frac{3^2 - 8}{3 - 2} = \frac{9 - 8}{1} = 1 \).

Thus, direct substitution can save time and effort, assuming no indeterminate form shows up.

Indeterminate forms often arise when evaluating functions, especially limits. Examples of indeterminate forms include \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). These forms indicate uncertainty, where standard arithmetic becomes unreliable.
To detect indeterminate forms, substitute the target value into the function's numerator and denominator separately. If both result in zero, for instance, you might face a \( \frac{0}{0} \) indeterminate form.
In this particular exercise, substituting \( x = 3 \) into our function does not yield an indeterminate form. Observing both the numerator \( x^2 - 8 = 1 \) and the denominator \( x - 2 = 1 \) at this point confirms that they do not both equate to zero, indicating we're free to use direct substitution without complication.

The Theorem on Limits makes it easier to calculate limits of rational functions. This theorem states that if the denominator of a rational function is non-zero at a particular value, the limit can be found by direct substitution.
In our provided exercise, the denominator \( x - 2 \) does not become zero at \( x = 3 \), allowing us to apply the theorem directly. Therefore, we substitute \( x = 3 \) into the function without worry and calculate:

• \( \lim_{x \to 3} \frac{x^2 - 8}{x - 2} = \frac{9 - 8}{1} = 1 \).

This demonstrates the usefulness of the theorem in confirming when and how we can safely use direct substitution to find limits without encountering undefined behavior.