Defining Limits and Using Limit Notation - AP Calculus AB Study Guide

The Core Idea: Defining Limits and Using Limit Notation

The concept of a limit is the foundational building block of calculus. It allows us to describe the behavior of a function, $f(x)$, as its input, $x$, gets extremely close to a particular value, which we'll call $c$. The central idea is not what happens at$x=c$, but rather what value the function's output, $f(x)$, is approaching as $x$ gets arbitrarily close to $c$ from both sides.

A limit exists if the function approaches a single, finite real number $L$ from both the left and the right of $c$. This concept is powerful because the function does not even need to be defined at $x=c$ for its limit to exist. The limit is concerned with the journey and the intended destination, not necessarily the arrival. We can investigate this behavior by examining a function's graph or a table of its values.

Key Definitions

The primary definition in this topic is that of the limit of a function.

The Limit of a Function

The limit of a function $f(x)$ as $x$ approaches $c$ is a real number $L$ if the values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ (but not equal to $c$).

Limit Notation

If the limit exists and is equal to the real number $L$, we express this using the following notation:

$$\underset{x\to c}{\lim }f(x)=L$$

This is read as "the limit of $f(x)$ as $x$ approaches $c$ equals $L$."

Understanding the Limit Concept

A critical distinction in calculus is the difference between the value of a function at a point, $f(c)$, and the limit of the function as $x$ approaches that point, ${\lim }_{x\to c}f(x)$.

• $f(c)$ is the actual output of the function when the input is exactly $c$. It is the $y$-coordinate of the point on the graph at $x=c$. This point may be a solid dot, or the function may be undefined at $c$.

• ${\lim }_{x\to c}f(x)$ is the $y$-value that the function's graph is "heading towards" as $x$ gets closer and closer to $c$ from both sides.

The limit can exist even if the function value is different or does not exist at all. Consider a graph with a "hole" at $(c,L)$ and a solid dot at $(c,M)$.

• The limit would be $L$, because as you trace the graph from both the left and right sides, you are heading towards the $y$-value of $L$.

• The function value would be $M$, because that is the actual, defined point at $x=c$.

• In this case, ${\lim }_{x\to c}f(x)\ne f(c)$.

Core Concepts & Rules

• The Approach: A limit describes the value a function $f(x)$ approaches as $x$ gets closer to a number $c$.

• Not the Actual Value: The limit is not necessarily the same as the function's value at $c$, $f(c)$.

• Existence: For a limit ${\lim }_{x\to c}f(x)$ to exist, the function must approach the same finite value $L$ as $x$ approaches $c$ from both the left and the right.

• Undefined Points: A function does not need to be defined at $x=c$ for the limit ${\lim }_{x\to c}f(x)$ to exist.

• Evidence for Limits: Limits can be determined or estimated by analyzing a graph of the function or by examining a table of values that shows function outputs for inputs very close to $c$.

• Proper Notation: Limits must be expressed using correct mathematical notation, such as ${\lim }_{x\to c}f(x)=L$.

Step-by-Step Example 1: Estimating a Limit from a Table

Problem: Estimate the value of ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$ by using a table of values.

Step 1: Identify the point of interest.

We are interested in the behavior of the function $f(x)=\frac{x^2-9}{x-3}$ as $x$ approaches $c=3$. Notice that plugging in $x=3$ directly results in $f(3)=\frac{9-9}{3-3}=\frac{0}{0}$, which is undefined. This means we must investigate the values near$x=3$.

Step 2: Create a table of values for $x$ approaching 3 from the left side.

Choose $x$-values that are slightly less than 3 and get progressively closer.

As $x$ approaches 3 from the left, $f(x)$ appears to be approaching 6.

Step 3: Create a table of values for $x$ approaching 3 from the right side.

Choose $x$-values that are slightly greater than 3 and get progressively closer.

As $x$ approaches 3 from the right, $f(x)$ also appears to be approaching 6.

Step 4: Conclude the limit value.

Since the function approaches the same value (6) from both the left and the right, we can estimate the limit.

Solution:${\lim }_{x\to 3}\frac{x^2-9}{x-3}=6$

Step-by-Step Example 2: Finding a Limit from a Graph

Problem: The graph of a function $g(x)$ is shown below. Use the graph to determine the values of (a) ${\lim }_{x\to -2}g(x)$ and (b) $g(-2)$.