91Ó°ÊÓ

Direct substitution is a method used to find the limit of a function as it approaches a particular point. In this process, we substitute the value which the variable is approaching directly into the function. If the function remains defined and produces a real number, we can say that the limit exists at that point.
For example, let's consider the function given in the exercise: \[ f(x) = \frac{3x^2 + x + 2}{x - 4} \]We need to find the limit as \( x \) approaches 0. Thus, we use direct substitution, replacing \( x \) by 0. Plugging it in, we calculate:

• Numerator: \( 3(0)^2 + 0 + 2 = 2 \)
• Denominator: \( 0 - 4 = -4 \)

The expression then simplifies to \( \frac{2}{-4} = -\frac{1}{2} \).
This result is a real number, confirming that the limit exists. Thus, direct substitution is not only a quick method but also effective in determining the existence of limits when conditions allow.

Evaluating limits is a fundamental concept for understanding the behavior of functions as variables approach specific values. It provides significant insight into the continuity and discontinuity of a function.
There are multiple techniques for evaluating limits:

• Direct Substitution: As discussed, you simply substitute the approaching value into the function.
• Factoring: Useful when substitution leads to an indeterminate form, like 0/0.
• Rationalization: Involves multiplying by a conjugate, often helpful in dealing with square roots.
• L'Hopital's Rule: Applies when you encounter fractions leading to indeterminate forms.

However, in our exercise, direct substitution was sufficient. Here, we could directly replace 0 in the function and achieve a concrete answer. It is crucial to first assess if direct substitution works, as it simplifies the evaluation greatly.
In other cases where direct substitution does not yield a valid answer, alternative methods should be explored.

Rational functions are fractions formed by polynomial expressions in the numerator and denominator. They are denoted generally as:\[ R(x) = \frac{p(x)}{q(x)} \]where \( p(x) \) and \( q(x) \) are polynomials.
An essential aspect of rational functions is their domain, which consists of all the input values except where the denominator \( q(x) \) equals zero.
In evaluating limits involving rational functions, it's common to encounter points where direct evaluation results in zero in the denominator. This signifies points of discontinuity or undefined values in the function.
For our function \( \frac{3x^2 + x + 2}{x - 4} \), when evaluating at \( x \) approaching 4, the denominator becomes zero, leading to a vertical asymptote at \( x = 4 \). However, when approaching any other value where the function remains defined, such as in the exercise with \( x \) approaching 0, direct substitution results in a valid limit.
Understanding rational functions is thus vital in analyzing behaviors like asymptotes and discontinuities, making them a key part of limit calculations.