91Ó°ÊÓ

An undefined function occurs when there's a value for which the given function does not return a finite result. In the context of the function presented in the exercise, which is \( F(x) = \frac{1}{x - 3} \), the function becomes undefined at \(x = 3\). This is because the denominator \( x - 3 \) equals zero when \( x = 3 \), and dividing by zero is undefined in mathematics. This means that as we approach the point where \( x = 3 \), our function fails to produce a meaningful value.

Here are key points to remember about undefined functions:

• An undefined function does not have a finite or usable value at certain points.
• We can often identify these points by looking for values that make the denominator zero in rational functions.
• In our specific case, \(x = 3\) is such a point where the function becomes undefined.

Understanding when and why a function is undefined helps us to better understand the behavior of the function near critical points.

The behavior of a function at a point, especially near undefined points, is crucial for understanding limits. For the function \( F(x) = \frac{1}{x - 3} \), the behavior as \( x \) approaches 3 is particularly interesting. As we get closer to \(x = 3 \) from the left (i.e., values just less than 3), \( x - 3 \) is a small negative number, making the function shoot off towards negative infinity: \( F(x) \rightarrow -\infty \).

Conversely, when approaching from the right (values just greater than 3), \( x - 3 \) is a small positive number, causing the function to head towards positive infinity: \( F(x) \rightarrow +\infty \). This divergent behavior on either side of \( x = 3 \) indicates that there's no single value the function approaches, and thus the limit does not exist at this point.

Remember these essential points when studying the behavior at a point:

• Analyze how the function behaves from both the left and right sides of the point in question.
• Notice if the function trends towards positive or negative infinity or if it approaches different finite values.
• Determining the behavior informs us whether the limit exists and, if so, what it is.

Continuous functions maintain a predictable behavior and do not have any breaks, holes, or jumps in their graph. For the second part of the exercise, the function \( F(x) = \frac{1}{x - 3} \) needs to be evaluated as \( x \) approaches 4. Unlike the behavior at \( x = 3 \), the function behaves normally at \( x = 4 \). Here, the denominator \( x - 3 \) equals 1, and the function produces a valid and finite result: \( F(4) = \frac{1}{4 - 3} = 1 \).

This means the function is defined and continuous at \( x = 4 \). As a result, the limit of \(F(x)\) as \(x\) approaches 4 is just the function's value at that point, which is 1.

Key highlights to grasp about continuous functions include:

• A continuous function has no breaks, holes, or jumps at the point of interest.
• For such functions, the limit at any given point is simply the function's value at that point.
• Understanding continuity helps in straightforwardly determining limits without the need for more detailed analysis.