Limits and One-Sided Limits in Calculus

Limits and One-Sided Limits

Definition of a Limit

The concept of a limit is fundamental in calculus and describes the behavior of a function as the input approaches a particular value. Formally, the limit of a function f(x) as x approaches a is written as:

• Limit Notation: $\lim_{x \to a} f(x) = L$

• This means that as x gets arbitrarily close to a (from either side), the values of f(x) approach L.

Example: $\lim_{x \to 4} f(x) = 3$ means as x approaches 4, f(x) approaches 3.

One-Sided Limits

Sometimes, it is useful to consider the behavior of a function as x approaches a value from only one side:

• Left-Hand Limit: $\lim_{x \to a^-} f(x)$ (as x approaches a from the left)

• Right-Hand Limit: $\lim_{x \to a^+} f(x)$ (as x approaches a from the right)

Key Point: The two-sided limit exists only if both one-sided limits exist and are equal:

• If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$, then $\lim_{x \to a} f(x) = L$.

• If the one-sided limits are not equal, the two-sided limit does not exist at that point.

Examples with Quadratic Function

Consider the function $f(x) = x^2$:

• From the left: $\lim_{x \to 2^-} f(x) = 4$

• From the right: $\lim_{x \to 2^+} f(x) = 4$

• From both sides: $\lim_{x \to 2} f(x) = 4$

This shows that the limit exists and equals 4 as x approaches 2.

Evaluating Limits from a Graph

To find limits using a graph, observe the value that f(x) approaches as x gets close to a specific point from the left and right. The actual value of f(x) at that point (if defined) is not always the same as the limit.

Example Analysis from the Graph:

• At x = 2:

  • Left-hand limit: Value f(x) approaches as x approaches 2 from the left.

  • Right-hand limit: Value f(x) approaches as x approaches 2 from the right.

  • If both are equal, the two-sided limit exists.

  • The filled or open circles indicate whether f(2) is defined and what its value is.

• At x = 4:

  • Repeat the process for left and right limits.

  • Compare with the actual value f(4) if defined.

Important: The limit as x approaches a value depends on the behavior near that value, not necessarily the function's value at that point.

Summary Table: Limits and Function Values

x-value	Left-Hand Limit	Right-Hand Limit	Two-Sided Limit	Function Value
2	Value from left	Value from right	Exists if left = right	f(2) (may differ)
4	Value from left	Value from right	Exists if left = right	f(4) (may differ)

Additional info: The table summarizes how to compare one-sided limits, two-sided limits, and function values at specific points using a graph.