Post 1: Introduction to Limits and Continuity in Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. Two fundamental concepts in calculus are limits and continuity. These concepts allow us to understand and analyze the behavior of functions, making them essential in various real-world contexts.

Limits:

Before diving into the concept of limits, let's start with the basic definition. A limit is the value that a function approaches as the input gets arbitrarily close to a particular point. We denote limits using the notation:

In this notation, $x$ is the input variable, $a$ is the specific point, and $f(x)$ is the function.

There are two types of limits:

• One-sided limits: These limits approach a specific point from either the left or the right side. The one-sided limit from the left is denoted as $\lim_{x \to a^-} f(x)$, and the one-sided limit from the right is denoted as $\lim_{x \to a^+} f(x)$.

• Two-sided limits: These limits consider the behavior of a function as the input approaches a specific point from both sides. The two-sided limit is denoted as $\lim_{x \to a} f(x)$, and for the limit to exist, the one-sided limits must be equal.

Example 1:

Let's consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. Find the limit of $f(x)$ as $x$ approaches 2.

To find the limit, we can simply evaluate the function at $x = 2$. However, this would result in an undefined value since the denominator becomes 0. Thus, we need to use algebraic manipulation to solve this problem.

First, we factorize the numerator:

Next, we cancel out the common factor of $(x - 2)$:

Now, we can directly substitute $x = 2$ into the simplified function:

Therefore, the limit of $f(x)$ as $x$ approaches 2 is 4.

Continuity:

Continuity is closely related to limits and refers to the behavior of a function without any abrupt changes or breaks. A function is considered continuous at a specific point $a$ if three conditions are satisfied:

1. The function is defined at $a$.
2. The limit of the function as $x$ approaches $a$ exists.
3. The limit of the function as $x$ approaches $a$ is equal to the value of the function at $a$.

If any of these conditions are not met, we have a discontinuity.

Example 2:

Consider the function $g(x) = \frac{x^2 - 4}{x - 2}$. Is the function continuous at $x = 2$?

From Example 1, we found that the limit of $g(x)$ as $x$ approaches 2 is 4. Now, let's evaluate the function directly at $x = 2$:

Since we obtained an indeterminate form, the function is not defined at $x = 2$. Therefore, the function $g(x)$ is not continuous at $x = 2$.

In conclusion, limits and continuity play a vital role in calculus, allowing us to understand and analyze the behavior of functions. They provide us with the tools to solve problems involving rates of change, derivatives, and identifying points of discontinuity, which are crucial in various real-world applications.