What are the Undefined Limits: Defining from Tables and Graphs

Let’s take a step-by-step approach for both scenarios:

1. Functions with Undefined Limits (from table):

Step 1: Understand the Question

• Typically, the question will be “Determine the limit of \( f(x) \) as \( x \) approaches \( c \) using the table.”

Step 2: Examine the Table

• Look at the table’s structure. You’ll typically have two columns: one for \( x \) values and one for \( f(x) \) values.

Step 3: Check for the Target x-value

• Identify the x-value \( c \) you’re approaching.

Step 4: Observe the Trends from Both Sides

• Look at the \( f(x) \) values as \( x \) approaches \( c \) from the left (values slightly less than \( c \)).
• Look at the \( f(x) \) values as \( x \) approaches \( c \) from the right (values slightly greater than \( c \)).

Step 5: Identify Inconsistencies

• If the \( f(x) \) values are not approaching the same value from both sides, then the limit is undefined.
• If the \( f(x) \) values oscillate or don’t approach any specific value, the limit may be undefined.

Step 6: State the Conclusion

• Clearly state whether the limit exists or is undefined based on your observations from the table.

Example:

Table for Function \( f(x) \):

As \( x \) approaches \(3\) from the left (values slightly less than \(3\)), \( f(x) \) seems to be approaching \(5\). However, as \( x \) approaches \(3\) from the right (values slightly greater than \(3\)), \( f(x) \) seems to be moving away from \(5\). Since the values of \( f(x) \) from the left and right of \(3\) aren’t converging to the same number, the limit of \( f(x) \) as \( x \) approaches \(3\) is undefined based on this table.

2. Functions with Undefined Limits (from graphs):

Step 1: Understand the Question

• Typically, the question will be “Determine the limit of \( f(x) \) as \( x \) approaches \( c \) using the graph.”

Step 2: Examine the Graph

• Look at the graph of \( f(x) \) to see its general behavior.

Step 3: Zoom in on the Target x-value

• Focus on the behavior of the graph near the point \( x = c \).

Step 4: Observe the Trends from Both Sides

• Examine the graph’s behavior as \( x \) approaches \( c \) from the left.
• Examine the graph’s behavior as \( x \) approaches \( c \) from the right.

Step 5: Identify Discrepancies

• If the graph approaches different y-values from the left and right of \( x = c \), the limit is undefined.
• Look for vertical asymptotes or points of discontinuity. If the graph goes to positive or negative infinity on one or both sides, the limit is undefined (or infinite).
• If the graph oscillates rapidly (like a sine wave that gets more frequent) and doesn’t settle on a value, the limit may be undefined.

Step 6: State the Conclusion

• Clearly state based on the graph’s behavior whether the limit is undefined or has a specific value.

Note: Remember that a function itself can be undefined at a point \( x = c \), but its limit as \( x \) approaches \( c \) might still exist. The key is to focus on the behavior of the function values as \( x \) gets closer and closer to \( c \), rather than the function value precisely at \( x = c \).

By following these steps, you’ll be able to determine whether a limit is undefined or not, whether you’re working with a table of values or a graph of the function.

Example:

Imagine the graph of a function that looks like a simple parabola \( y = x^2 \) everywhere except at \( x = 0 \). At \( x = 0 \), instead of continuing smoothly, there’s a “hole” in the graph (a point of discontinuity), and a separate point plotted at \((0,2\)).

If you approach \( x = 0 \) from the left or the right, the y-values seem to be approaching \(0\) (because of the parabolic shape). However, the actual value of the function at \( x = 0 \) is given as \(2\). Even though there’s a defined value at \( x = 0 \), the limits from the left and right as \( x \) approaches \(0\) are both \(0\). The limit is well-defined, but the function value at that point is different from the limit, illustrating a removable discontinuity.

In other cases, such as with vertical asymptotes, the function might shoot off toward positive or negative infinity as it approaches a particular x-value. For instance, the function \( f(x) = \frac{1}{x} \) has a vertical asymptote at \( x = 0 \). As you approach \(0\) from the left, the function dives down to negative infinity, and as you approach from the right, it shoots up to positive infinity. In this scenario, the limits from both sides are undefined, and so is the overall limit at \( x = 0 \).