Introduction to Limits

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

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◇Overview

The concept of a limit is fundamental to calculus. It describes the value that a function approaches as the input approaches a certain value.

If $f(x)$ becomes arbitrarily close to a unique number $L$ as $x$ approaches $c$ from both sides, then we say that the limit of $f(x)$ as $x$ approaches $c$ is $L$. This is written as:
$\lim_{x \to c} f(x) = L$

Limits can be investigated using numerical methods (tables of values) or graphical methods (looking at the graph of $y=f(x)$).

Key Considerations: The function does not necessarily need to be defined at $x = c$ for the limit to exist. For a limit to exist at a point, the left-hand limit and right-hand limit must exist and be equal. If the function approaches different values from the left and right, the limit does not exist (DNE). Limits that lead to division by zero (e.g., evaluating $\frac{1}{0}$) often indicate vertical asymptotes or infinite limits.

◇Important Formulas

General Limit Notation:
$\lim_{x \to c} f(x) = L$ Left-Hand Limit:
$\lim_{x \to c^-} f(x) = L$ Right-Hand Limit:
$\lim_{x \to c^+} f(x) = L$ Existence of Limit:
$\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$.

◇Worked Examples

Example 1: Investigating a limit using a table of values
Consider $\lim_{x \to 2} (1 + 3x)$.
Approaching $x=2$ from the left (e.g., $1.9, 1.99, 1.999$) and from the right (e.g., $2.1, 2.01, 2.001$), we observe the values of $f(x)$:
As $x \to 2^-$, $f(x) \to 7$.
As $x \to 2^+$, $f(x) \to 7$.
Since both approach 7, we conclude $\lim_{x \to 2} (1 + 3x) = 7$.

Example 2: A limit that Does Not Exist (DNE)
Investigate $\lim_{x \to 4} f(x)$ where $f(x)$ approaches 5 from the left and approaches 3 from the right.
Since the left-hand limit is $5$ and the right-hand limit is $3$, they are not equal.
Therefore, the limit does not exist (DNE).
$\lim_{x \to 4^-} f(x) = 5$
$\lim_{x \to 4^+} f(x) = 3$
$\lim_{x \to 4} f(x) \text{ DNE}$

◇References & Navigation

• Previous Topic: ◇Functions and Their Graphs
• Next Topic: ◇Definition of the Limit and Limit Theorems
• Related Notes: ◇One-Sided Limits and Infinite Limits, ◇Limit at Infinity