Limits intro

Limits intro

"Limits intro" typically refers to the foundational concepts in calculus that describe how a function behaves as its input approaches a certain value. Here are the key concepts:

Definition of a Limit: A limit examines the value that a function approaches as the input approaches a specific point. For instance, as $x$ gets closer to $a$ , we look at what $f(x)$ approaches.
One-Sided Limits: Limits can be approached from the left (denoted as $\lim_{x \to a^-} f(x)$ ) or from the right (denoted as $\lim_{x \to a^+} f(x)$ ).
Existence of Limits: A limit exists if both one-sided limits are equal. If they are not, the limit does not exist.
Infinite Limits: These describe behavior where a function approaches infinity (or negative infinity) as $x$ approaches a certain value, indicating vertical asymptotes.
Limits at Infinity: Limits can also consider what happens as $x$ approaches infinity, focusing on the long-term behavior of a function.

Understanding limits is crucial for later topics in calculus, such as derivatives and integrals.

In this article

Part 1: Limits intro

In this video, we learn about limits, a fundamental concept in calculus. Limits help us understand what a function approaches as the input gets closer to a certain value, even when the function is undefined at that point. The video demonstrates this concept using two examples with different functions.

Here are the key points to learn when studying "Limits Intro":

Definition of Limits: Understand that a limit describes the behavior of a function as it approaches a specific point from either direction.
Notation: Familiarize yourself with limit notation, such as $\lim_{x \to c} f(x)$ , which indicates the limit of $f(x)$ as $x$ approaches $c$ .
One-Sided Limits: Learn the concept of one-sided limits, denoted as $\lim_{x \to c^-} f(x)$ (from the left) and $\lim_{x \to c^+} f(x)$ (from the right).
Existence of Limits: A limit exists only if both one-sided limits are equal.
Limit Laws: Get acquainted with basic rules for calculating limits, including the sum, product, and quotient laws.
Limits involving Infinity: Understand limits where $x$ approaches infinity or negative infinity, and what it means for functions to approach horizontal asymptotes.
Continuous Functions: Learn the relationship between limits and continuity; specifically, that if a function is continuous at a point, the limit at that point equals the function's value.
Indeterminate Forms: Recognize situations that lead to indeterminate forms (like $0/0$ ) and know methods to resolve them (e.g., factoring, rationalizing, L'Hôpital's Rule).
Graphical Interpretation: Practice analyzing graphs to visually identify limits, especially around discontinuities and asymptotes.

By focusing on these key points, you'll have a foundational understanding of limits in calculus.

Part 1: Limits intro

Related topics

Estimating limits from graphs

Estimating limits from tables

Formal definition of limits (epsilon-delta)

Properties of limits

Limits by direct substitution

Limits using algebraic manipulation

Strategy in finding limits

Squeeze theorem

Types of discontinuities

Continuity at a point

Continuity over an interval

Removing discontinuities

Infinite limits

Limits at infinity

Intermediate value theorem