Question

Describe three situations where \(\lim f(x)\) does not exist.

Short answer

Limits do not exist when one-sided limits differ, when behavior is unbounded, or when the function oscillates.

Step-by-step solution

Identify the Concept of a Limit

A limit \(\lim_{x \to a} f(x)\) does not exist at a point \(x = a\) if the function approaches different values from the left and from the right, or if the function does not approach any value as \(x\) approaches \(a\). Let's explore specific situations where limits do not exist.

Consider Non-Equal One-Sided Limits

If \(\lim_{x \to a^-} f(x) eq \lim_{x \to a^+} f(x)\), the limit does not exist at \(x = a\). An example of this is the function \(f(x) = \frac{|x|}{x}\), where at \(x = 0\), the left-hand limit is \(-1\) and the right-hand limit is \(1\), so the limit does not exist.

Consider Unbounded Behavior

When a function approaches infinity or negative infinity as \(x\) approaches \(a\), the limit does not exist. For instance, \(f(x) = \frac{1}{x}\) as \(x \to 0\) from either side becomes infinitely large, indicating the limit is not finite.

Consider Oscillatory Behavior

Another situation where the limit does not exist is when the function oscillates infinitely between fixed values as \(x\) approaches \(a\). For example, \(f(x) = \sin(\frac{1}{x})\) has no limit as \(x \to 0\) because the sine function oscillates between -1 and 1 infinitely.

Key concepts

One-Sided Limits

One-sided limits occur when we assess the behavior of a function as it approaches a specific point from one direction—either from the left or the right. This is valuable in understanding how a function behaves at a point, especially when both sides may not meet.
The notation for a left-hand limit is \( \lim_{x \to a^-} f(x) \), which reflects approaching the point \( a \) from the left. Conversely, \( \lim_{x \to a^+} f(x) \) denotes a right-hand limit, approaching from the right.
If these one-sided limits are not equal, it can point to certain interesting behaviors, such as a jump discontinuity. For instance, the function \( f(x) = \frac{|x|}{x} \) has differing left-hand and right-hand limits at \( x = 0 \). Here, the left-hand limit is \(-1\), and the right-hand limit is \(1\). Since \( -1 eq 1 \), the overall limit \( \lim_{x \to 0} f(x) \) does not exist.
In cases where one-sided limits differ, it signals that the function behaves differently from each direction, meaning the function's behavior is not smoothly transitioning at that point.

Unbounded Behavior

Unbounded behavior refers to situations where a function grows indefinitely to positive or negative infinity as it approaches a certain point. This typically happens near vertical asymptotes on graphs and signals that the limit does not exist because the function never settles to a specific value.
Take the function \( f(x) = \frac{1}{x} \) as an example. As \( x \) approaches \( 0 \) from the right, the value of \( \frac{1}{x} \) shoots overhead to infinity.
On the other hand, when \( x \) approaches \( 0 \) from the left, \( f(x) = \frac{1}{x} \) descends into negative infinity. The limits coming from either direction are unbounded, hence \( \lim_{x \to 0} f(x) \) is not existent.
In mathematical analysis, encountering unbounded behavior suggests the function's values keep becoming larger or smaller without ever stopping, which is why a single limit cannot be determined.

Oscillatory Behavior

Oscillatory behavior involves rapid fluctuations of a function between set values as it nears a particular point, preventing the establishment of a limit. This is typically the case in periodic functions that speed up their oscillation near certain points.
An illustrative example is \( f(x) = \sin\left(\frac{1}{x}\right) \), which showcases no defined limit as \( x \to 0 \). Why is this? Well, because \( \sin \left(\frac{1}{x}\right) \) swiftly oscillates between \(-1\) and \(1\) as \( x \) comes closer and closer to \(0\), never locking onto a single value.
In essence, oscillatory behavior causes the function's value to swing back and forth so intensely near the point of interest that a single limit cannot exist.
Such functions may encounter pending double-ended oscillations and predominant error when solving critical mathematical problems requiring finely-tuned precision.