Section 2-2: Understanding Limits

Welcome to Section 2-2 of Calculus 1! In this section, we'll be exploring the fundamental concept of limits. Limits are the building blocks of calculus, so understanding them well is crucial for your success. Let's get started!

Intuitive Definition of a Limit

The idea behind a limit is to describe the behavior of a function $f(x)$ as $x$ approaches a specific value $a$. Think of it as asking: "What value does $f(x)$ get close to when $x$ is really, really close to $a$?"

Formally, we write:

$$ \lim_{x \to a} f(x) = L $$

This reads as "the limit of $f(x)$ as $x$ approaches $a$ equals $L$." It means that we can make the values of $f(x)$ arbitrarily close to $L$ by restricting $x$ to be sufficiently close to $a$, but not equal to $a$. It is important to remember we are concerned with the values *around* $a$, not necessarily *at* $a$.

Example: Consider the function $f(x) = x + 3$. What happens as $x$ approaches 4?

$$ \lim_{x \to 4} (x + 3) = 7 $$

As $x$ gets closer and closer to 4, $f(x)$ gets closer and closer to 7.

One-Sided Limits

Sometimes, we need to consider what happens as $x$ approaches $a$ from only one side. This leads to the concept of one-sided limits:

  • Left-Hand Limit: The limit of $f(x)$ as $x$ approaches $a$ from the left (i.e., $x$ is less than $a$):
$$ \lim_{x \to a^-} f(x) = L $$
  • Right-Hand Limit: The limit of $f(x)$ as $x$ approaches $a$ from the right (i.e., $x$ is greater than $a$):
$$ \lim_{x \to a^+} f(x) = L $$

Key Point: For the regular limit $\lim_{x \to a} f(x)$ to exist, both the left-hand limit and the right-hand limit must exist and be equal. That is, $$ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L $$.

Infinite Limits

What happens when the values of $f(x)$ become arbitrarily large (positive or negative) as $x$ approaches $a$? This is where infinite limits come in. We write:

$$ \lim_{x \to a} f(x) = \infty $$

or

$$ \lim_{x \to a} f(x) = -\infty $$

This means that the values of $f(x)$ can be made arbitrarily large (as large as we please) by taking $x$ sufficiently close to $a$, but not equal to $a$.

Vertical Asymptotes: If either $\lim_{x \to a^-} f(x) = \pm \infty$ or $\lim_{x \to a^+} f(x) = \pm \infty$, then the line $x = a$ is a vertical asymptote of the curve $y = f(x)$.

Understanding these concepts of limits, one-sided limits and infinite limits will give you the tools needed to be successful in calculus. Keep practicing, and you'll master them in no time!