Lesson: Understanding Limits and Continuity

Overview

Limits and continuity are fundamental ideas in calculus. They help us understand how functions behave, especially near points where they might not be well-defined. Mastering these concepts is essential before learning about derivatives (differentiation).

1. What is a Limit?

A limit describes the value that a function approaches as the input (usually $x$) gets closer and closer to a certain point.

Example:

Suppose $f(x) = 2x$. What is the limit as $x$ approaches 3?

$$ \lim_{x \to 3} 2x = 2 \times 3 = 6 $$

This means as $x$ gets closer to 3, $f(x)$ gets closer to 6.

2. Why Are Limits Important?

Limits help us:

Understand function behavior near points where the function might not be defined
Form the basis for defining derivatives (slopes of curves)

3. Graphical Example

Consider the function:

$$ f(x) = \begin{cases} x^2 & \text{if } x \neq 2 \ 5 & \text{if } x = 2 \end{cases} $$

For $x \neq 2$, $f(x) = x^2$
At $x = 2$, $f(2) = 5$

Question: What is $\lim_{x \to 2} f(x)$?

Solution:

As $x$ gets close to 2 (but not equal to 2), $f(x)$ gets close to $2^2 = 4$.
So, $\lim_{x \to 2} f(x) = 4$, even though $f(2) = 5$.

Key Point: The limit depends on nearby values, not just the value at the point.

4. Continuity

A function is continuous at a point $x = a$ if:

$f(a)$ is defined
$\lim_{x \to a} f(x)$ exists
$\lim_{x \to a} f(x) = f(a)$

If any of these are not true, the function is discontinuous at $x = a$.

Example of Discontinuity

Using the previous function:

$f(2) = 5$
$\lim_{x \to 2} f(x) = 4$

Since $f(2) \neq \lim_{x \to 2} f(x)$, $f(x)$ is not continuous at $x = 2$.

5. Common Misconceptions

Undefined Point ≠ No Limit: A function can have a limit at a point where it is not defined.
Jump Discontinuity: If the left and right limits are not equal, the limit does not exist at that point.

Example:

$$ g(x) = \begin{cases} 1 & \text{if } x < 0 \ 2 & \text{if } x \geq 0 \end{cases} $$

$\lim_{x \to 0^-} g(x) = 1$
$\lim_{x \to 0^+} g(x) = 2$
The overall limit $\lim_{x \to 0} g(x)$ does not exist because the left and right limits are different.

6. Practice Questions

Find $\lim_{x \to 1} (3x + 2)$
Is the function $h(x) = \frac{x^2 - 1}{x - 1}$ continuous at $x = 1$? (Hint: Simplify first)
Sketch a graph of a function with a jump discontinuity at $x = 0$.

Summary

Limits describe how functions behave near a point.
Continuity means the function has no breaks, jumps, or holes at a point.
Limits can exist even if the function is not defined at that point.
Understanding these ideas is crucial for learning about derivatives and calculus.