Limits intro

Limits intro

The introduction to limits in calculus focuses on understanding the behavior of functions as they approach a particular point or value. Here are some key concepts:

Definition of a Limit: A limit describes the value that a function approaches as the input approaches a specific point. It can be expressed as $\lim_{x \to a} f(x) = L$ , meaning as $x$ gets closer to $a$ , $f(x)$ approaches $L$ .
Left-hand and Right-hand Limits: Limits can be examined from two directions:
- Left-hand limit: Approaching from the left (denoted as $\lim_{x \to a^-} f(x)$ ).
- Right-hand limit: Approaching from the right (denoted as $\lim_{x \to a^+} f(x)$ ). For a limit to exist, both must be equal.
Existence of Limits: A limit exists if both the left-hand and right-hand limits are equal. If they are not, or if the function behaves erratically, the limit may not exist.
Infinite Limits: If $f(x)$ increases or decreases without bound as $x$ approaches a certain value, we denote this as $\lim_{x \to a} f(x) = \infty$ or $-\infty$ .
Special Cases: Some limits involve indeterminate forms (such as $\frac{0}{0}$ ), which may require further techniques (like L'Hôpital's Rule) to evaluate.
Continuity: A function is continuous at a point $a$ if $\lim_{x \to a} f(x) = f(a)$ . This means the limit exists and equals the function value at that point.

Overall, limits lay the foundation for understanding derivatives and integrals in calculus.

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 focus on when studying "Limits Intro":

Definition of Limits: Understand the basic concept of a limit as a value that a function approaches as the input approaches a certain point.
Notation: Familiarize yourself with the notation $\lim_{x \to a} f(x) = L$ , which means that as $x$ approaches $a$ , $f(x)$ approaches $L$ .
One-Sided Limits: Learn the difference between left-hand limits $\lim_{x \to a^-} f(x)$ and right-hand limits $\lim_{x \to a^+} f(x)$ .
Limit Existence: A limit exists only if both the left and right limits are equal.
Evaluating Limits: Explore techniques for evaluating limits, including direct substitution, factoring, and rationalizing.
Limits at Infinity: Understand how to evaluate limits where $x$ approaches positive or negative infinity.
Continuous Functions: Recognize that limits and continuity are closely related; a function is continuous at a point if the limit exists and equals the function's value at that point.
Limit Theorems: Familiarize yourself with fundamental limit theorems, such as the sum, product, and quotient rules.
Indeterminate Forms: Identify indeterminate forms like $\frac{0}{0}$ and $\frac{\infty}{\infty}$ , and learn techniques like L'Hôpital's Rule to resolve them.
Applications of Limits: Understand the role of limits in calculus, including derivatives and integrals.

Focusing on these key points will provide a solid foundation for understanding 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