Question

What are the three ways in which a limit may fail to exist?

Short answer

Limits may fail to exist due to infinite oscillation, unbounded behavior, or differing left and right limits.

Step-by-step solution

Identify Conceptual Overview

Limits in calculus help us understand the value that a function approaches as the input approaches a certain point. Sometimes, limits do not exist, and there are three main types of failures.

Understand Infinite Oscillation

A limit does not exist when a function oscillates infinitely as the input approaches a certain point. An example is the function \( f(x) = \sin\left(\frac{1}{x}\right) \) as \( x \) approaches 0; the function keeps oscillating between 1 and -1, never settling towards a single value.

Recognize Unbounded Behavior

Another way a limit can fail to exist is when the function becomes unbounded, meaning it goes to infinity or negative infinity as the input approaches a certain value. For example, \( \lim_{{x \to 0}} \frac{1}{x^2} = \infty \), indicating the function heads towards infinity and thus the limit does not exist.

Different Left and Right Limits

A limit does not exist when the left-hand limit is different from the right-hand limit. For instance, with the piecewise function \( f(x) = \begin{cases} x^2 & \text{if } x < 0 \ x + 1 & \text{if } x \geq 0 \end{cases} \), approaching from left yields a limit of 0, and from the right yields a limit of 1 at \( x = 0 \).

Key concepts

Infinite Oscillation

In calculus, a limit might not exist when a function experiences what's called infinite oscillation. This happens when a function doesn't settle at any single value as the input gets close to a certain point. The function continually jumps between values, like a spinning top that never stops.
An illustrative example of this concept is the function \( f(x) = \sin\left(\frac{1}{x}\right) \) as \( x \) approaches zero. As \( x \) gets closer to 0, the value of \( \frac{1}{x} \) becomes very large, leading to the sine function oscillating wildly between 1 and -1, infinitely and rapidly.

• Such oscillations prevent the function from approaching any particular value.
• The limit does not exist as there is no predictable outcome.

Understanding infinite oscillation is crucial because it highlights a situation where a function's behavior becomes erratic and unpredictable, making a limit inexistent.

Unbounded Behavior

Unbounded behavior is another significant reason why a limit may fail to exist in calculus. When a function's value becomes exceedingly large, either positive or negative, as it approaches a specific point, it's said to be unbounded. This means that the function is racing off to infinity or negative infinity and never settles to a finite number.
Take for instance the expression \( \lim_{{x \to 0}} \frac{1}{x^2} \). As \( x \) gets closer to zero, \( \frac{1}{x^2} \) skyrockets to positive infinity. The values increase without bound, showing unbounded behavior.

• This limitless growth means there's no specific number the function is tending towards.
• In such cases, the limit does not exist due to the function's runaway nature.

Grasping this concept is vital as it showcases a scenario where a function's growth towards infinity precludes the existence of a valid limit.

Left and Right Limits

One more way a limit may not exist is when there is a discrepancy between left and right limits. A limit only exists if both these directional limits meet at the same endpoint. If they differ, no single limit exists for that point.
Consider the piecewise function \( f(x) = \begin{cases} x^2 & \text{if } x < 0 \ x + 1 & \text{if } x \geq 0 \end{cases} \). As \( x \) approaches zero, coming from the left side (\( x < 0 \)), the function heads towards 0, whereas from the right side (\( x \geq 0 \)), it approaches 1.

• The left-hand limit results in 0, while the right-hand limit results in 1.
• Since these limits do not match, the overall limit at \( x = 0 \) does not exist.

Understanding different left and right limits is essential as it points out the need for consistency in directional limits to ensure a limit is well-defined.