Limits and Continuity

Introduction to Limits

The concept of a limit is fundamental in calculus and helps us understand the behavior of a function as its input approaches a specific value. Limits are used to study discontinuities and deal with undefined values. They form the foundation for differentiation and integration.

This equation means that as $x$ approaches $c$, the function $f(x)$ gets arbitrarily close to $L$.

Definition of a Limit

Intuitive Understanding

Imagine you're walking towards a point on a map. A limit describes what happens to your function (like elevation) as you get closer and closer to that point, but not exactly at it.

Formal Definition (ε-δ Definition)

The formal definition of a limit is more precise and involves the ε-δ (epsilon-delta) concept:

In simpler terms, this means for any small positive distance $\epsilon$ we choose, we can find a small positive distance $\delta$ around the point $c$ such that the function values stay within $\epsilon$ of the limit $L$.

Properties of Limits

Limits have several important properties that make them powerful mathematical tools:

Basic Limit Rules

• Sum Rule: $\lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$
• Difference Rule: $\lim_{x \to c} (f(x) - g(x)) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$
• Product Rule: $\lim_{x \to c} (f(x) \cdot g(x)) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
• Quotient Rule: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$(provided $\lim_{x \to c} g(x) \neq 0$)

One Sided Limits

One-sided limits examine the behavior of a function as it approaches a point from either the left or right side.

• Left-Hand Limit: $\lim_{x \to c^-} f(x)$ approaches the point from values less than $c$
• Right-Hand Limit: $\lim_{x \to c^+} f(x)$ approaches the point from values greater than $c$

A limit exists if and only if both left-hand and right-hand limits exist and are equal.

Limits at Infinity and Infinite Limits

Horizontal and Vertical Asymptotes

Asymptotes describe the long-term behavior of functions:

• Horizontal Asymptote: $\lim_{x \to \pm\infty} f(x) = L$Describes the function's behavior as $x$ approaches positive or negative infinity.
• Vertical Asymptote: Occurs when $\lim_{x \to c} f(x) = \pm\infty$Typically happens when the denominator of a rational function approaches zero.

Example of a horizontal asymptote:

Continuity

Definition of Continuity

A function is continuous at a point if:

1. The function is defined at that point
2. The limit exists at that point
3. The limit equals the function's value at that point

Types of Discontinuities

• Removable Discontinuity: A "hole" in the function that can be filled by redefining the function at a single point.
• Jump Discontinuity: The function has different left-hand and right-hand limits.
• Infinite Discontinuity: The limit approaches infinity, often seen in rational functions.

The Intermediate Value Theorem

A powerful theorem that applies to continuous functions:

In simpler terms, if a continuous function takes on two different values, it must take on all values between those two values at some point in the interval.

Practical Example

Imagine a temperature function $f(t)$ continuous over a day. If the temperature is 20°C in the morning and 30°C in the evening, the theorem guarantees that at some point during the day, the temperature will be exactly 25°C.

Real-Life Applications

Limits and continuity have numerous real-life applications, such as:

• Physics: Calculating instantaneous velocity or acceleration by taking limits.
• Economics: Determining marginal cost and revenue functions.
• Engineering: Analyzing system stability and performance by studying continuous models.
• Computer Graphics: Creating smooth animations and transitions.

Example

In traffic flow analysis, limits are used to measure the instantaneous rate of vehicles passing a checkpoint by analyzing data over an increasingly small interval of time.