Problem 55 - Precalculus

Question

What does it mean to say the limit of \(f(x),\) as \(x\) approaches \(c,\) is undefined?

Step-by-Step Solution

Answer

A limit is undefined if \( f(x) \) doesn't approach a specific value near \( c \).

Step 1: Understanding Limits

The statement `the limit of \( f(x) \) as \( x \) approaches \( c \) is undefined` suggests that as \( x \) gets closer to the value \( c \), \( f(x) \) does not approach any particular value. We want to explore why this might happen.

Step 2: Reasons for an Undefined Limit

A limit can be undefined for several reasons. For instance, \( f(x) \) may have one-side limits that are different, \( f(x) \) may oscillate indefinitely as \( x \) approaches \( c \) (like \( \sin \left( \frac{1}{x-c} \right) \)), or \( f(x) \) may approach infinity or negative infinity.

Step 3: Identifying One-Sided Limits

One scenario where a limit is undefined is when the left-hand limit \( \lim_{{x \to c^-}} f(x) \) and the right-hand limit \( \lim_{{x \to c^+}} f(x) \) exist but are not equal. This means the function approaches different values from either side of \( c \).

Step 4: Considering Oscillation

Another case is when \( f(x) \) oscillates between values as \( x \) approaches \( c \). An example is \( f(x) = \sin \left( \frac{1}{x-c} \right) \), where \( f(x) \) does not settle toward any single value as \( x \) nears \( c \).

Step 5: Infinite Behavior

Finally, if \( f(x) \) increases or decreases without bound as \( x \) approaches \( c \), the limit is also undefined. For instance, \( 1/(x-c)^2 \) approaches infinity as \( x \) approaches \( c \).

Step 6: Conclusion on Undefined Limits

In summary, a limit is undefined when \( f(x) \) does not approach a specific finite value as \( x \) approaches \( c \), due to differing one-side limits, oscillations, or infinite behavior.

Key Concepts

One-Sided Limits

In calculus, a measure of a function's limit at a point is often determined by examining how the function behaves as it approaches the point from either side. This is known as checking the one-sided limits.

• The left-hand limit is denoted as \( \lim_{{x \to c^-}} f(x) \) and considers values of \( x \) that are less than \( c \) but getting closer to it.
• The right-hand limit is represented by \( \lim_{{x \to c^+}} f(x) \), taking into account \( x \) values that are greater than \( c \) and approaching it.

A limit is defined at a point if both one-sided limits exist and are equal. However, if these limits differ, the overall limit at that point is undefined. A common situation where this occurs is with piecewise functions that have different rules for \( x \) values less than or greater than \( c \).
It's important to analyze both directions to fully understand a function's behavior near a specific point.

Oscillation

Oscillation refers to the repeated and continuous fluctuation of values around a mean or between two points—even as the input approaches a certain value. When dealing with limits, oscillation can lead to the limit being undefined, as the function doesn't settle on a steady value.
An easy-to-understand example is the function \( f(x) = \sin \left( \frac{1}{x-c} \right) \). As \( x \) approaches \( c \), the values of \( \left( \frac{1}{x-c} \right) \) oscillate more and more rapidly, causing the sine function to swing back and forth between -1 and 1 indefinitely. The key aspect of oscillation in the context of limits is that the output values never stabilize to a single finite number.
Understanding oscillation is crucial because it explains why some limits are not defined even though the individual function values remain finite. The constant back-and-forth change prevents the determination of a single limit value.

Infinite Behavior

When a function exhibits infinite behavior as \( x \) approaches \( c \), it means that the function values grow larger and larger in magnitude without ever stabilizing. In other words, they "shoot off" to infinity or negative infinity, making the limit undefined at that point.
Common functions that exhibit infinite behavior include rational functions like \( f(x) = \frac{1}{(x-c)^2} \). As \( x \) nears \( c \), the denominator approaches zero, causing the value of the function to skyrocket towards infinity. This behavior further supports why the limit cannot be characterized as a specific number.
In cases of infinite behavior, it's essential to note the direction of approach—whether the function heads towards positive or negative infinity. This knowledge helps in understanding the function’s asymptotic behavior and in outlining its limits more accurately.