Definition of Limits and One-Sided Limits 2.2

Definition of Limits

Introduction to Limits

The concept of a limit is fundamental in calculus and describes the behavior of a function as its input approaches a particular value. Limits are used to define continuity, derivatives, and integrals.

• Limit Notation: The limit of a function $f(x)$ as $x$ approaches $a$ is written as $\lim_{x \to a} f(x) = L$.

• Interpretation: This means that as $x$ gets arbitrarily close to $a$, the value of $f(x)$ approaches $L$.

• Graphical Representation: On a graph, the limit describes the $y$-value that the function approaches as $x$ nears $a$, regardless of the actual value of $f(a)$.

Example:

• If $\lim_{x \to 2} f(x) = 4$, then as $x$ approaches 2, $f(x)$ gets closer to 4.

• It is possible that $f(2)$ is not equal to 4, or even that $f(2)$ is undefined.

Evaluating Limits from Graphs

Limits can often be estimated or determined by examining the graph of a function near the point of interest.

• Function Value vs. Limit: The value of the function at a point ($f(a)$) may differ from the limit as $x$ approaches that point.

• Does Not Exist (DNE): If the function does not approach a single value as $x$ approaches $a$, the limit is said to not exist (DNE).

Example:

• Given $f(2) = \text{DNE}$ and $\lim_{x \to 2} f(x) = 4$, the function is not defined at $x=2$, but the limit exists and equals 4.

• Given $g(2) = 6$ and $\lim_{x \to 2} g(x) = 4$, the function value and the limit are different.

One-Sided Limits

Left-Sided and Right-Sided Limits

Sometimes, the behavior of a function as $x$ approaches a point from the left differs from its behavior as $x$ approaches from the right. These are called one-sided limits.

• Left-Sided Limit: $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$.

• Right-Sided Limit: $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$.

• If both one-sided limits exist and are equal, the (two-sided) limit exists and equals that value.

• If the one-sided limits are not equal, the (two-sided) limit does not exist (DNE).

Example:

• Given $f(2) = 1$, $\lim_{x \to 2^-} f(x) = 4$, $\lim_{x \to 2^+} f(x) = 1$, and $\lim_{x \to 2} f(x) = \text{DNE}$, the left and right limits are not equal, so the overall limit does not exist.

Practice: Reading Limits from Graphs

Examples with Piecewise and Discontinuous Functions

Below are examples of evaluating function values and limits from graphs. These illustrate how limits can exist even when the function is undefined or discontinuous at a point.

Expression	Value
$g(-1)$	3
$\lim_{x \to -1} g(x)$	2
$\lim_{x \to 1} g(x)$	2
$\lim_{x \to 1^-} g(x)$	2
$g(1)$	2
$\lim_{x \to 1^+} g(x)$	DNE
$\lim_{x \to 3} g(x)$	4
$\lim_{x \to 5} g(x)$	5

Expression	Value
$g(2)$	3
$\lim_{x \to 2^-} g(x)$	2
$\lim_{x \to 2^+} g(x)$	3
$\lim_{x \to 2} g(x)$	DNE
$g(3)$	2
$\lim_{x \to 3} g(x)$	3
$\lim_{x \to 4} g(x)$	DNE
$\lim_{x \to 5} g(x)$	2

Key Points to Remember

• Limits describe approaching behavior, not necessarily the value at the point.

• One-sided limits are useful for analyzing discontinuities and jumps in functions.

• DNE (Does Not Exist) is used when the function does not approach a single value from both sides.

Summary Table: Limit Terminology

Term	Meaning
Limit	Approaching value as $x$ nears $a$
One-sided limit	Approaching value from left ($a^-$) or right ($a^+$)
DNE	Limit does not exist
Function value	Actual value at $x=a$

Additional info: The notes are based on lecture slides and include graphical examples, definitions, and practice problems for evaluating limits and one-sided limits from graphs.