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

The Core Idea: Defining Limits and Using Limit Notation

The concept of a limit is the foundational building block of calculus. It provides a precise way to describe how a function behaves as its input gets closer and closer to a particular value. The core idea is not about what happens at the value, but rather what value the function's output approaches as the input gets arbitrarily close to that value from both sides.

If a function's output, $f(x)$, can be made as close as we want to a single real number $L$ just by choosing an input $x$ that is sufficiently close to a number $c$, we say that the limit of $f(x)$ as $x$ approaches $c$ is $L$. This concept allows us to analyze a function's behavior at points of interest, such as where it might be undefined, and forms the basis for defining continuity, derivatives, and integrals. When a function approaches such a finite value $L$, we say the limit exists and the function converges to $L$.

Key Definitions and Notation

The language of limits is expressed through specific mathematical notation. Understanding this notation is essential for both interpreting and solving problems in calculus.

The Two-Sided Limit

The statement "the limit of $f(x)$ as $x$ approaches $c$ is $L$" is written as:

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

• $\lim$ is the limit operator.

• $x\to c$ indicates that $x$ is approaching the value $c$.

• $f(x)$ is the function being examined.

• $L$ is the real number that $f(x)$ approaches.

One-Sided Limits

We can also describe the behavior of a function as it approaches $c$ from only one side.

1. The Left-Hand Limit: This describes the value the function approaches as $x$ gets close to $c$ from values less than$c$. The notation is:

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

   The superscript "$-$" on $c$ indicates the approach is from the left side (the negative side) on the number line.

2. The Right-Hand Limit: This describes the value the function approaches as $x$ gets close to $c$ from values greater than$c$. The notation is:

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

   The superscript "$+$" on $c$ indicates the approach is from the right side (the positive side) on the number line.

Understanding the Condition for Existence

A common point of confusion is the difference between the value of a function at a point, $f(c)$, and the limit of the function as $x$ approaches $c$. The limit is concerned only with the values of $f(x)$ for $x$near$c$, not the value of $f(x)$at$x=c$. A function does not even need to be defined at $x=c$ for the limit to exist.

The most critical rule for the existence of a two-sided limit is based on the behavior of the one-sided limits.

The Condition for a Limit to Exist:

The two-sided limit ${\lim }_{x\to c}f(x)$ exists and is equal to a real number $L$ if and only if the left-hand limit and the right-hand limit both exist and are equal to $L$.

Symbolically:

$$\underset{x\to c}{\lim }f(x)=L⟺\underset{x\to c^{-}}{\lim }f(x)=L\text{and}\underset{x\to c^{+}}{\lim }f(x)=L$$

If ${\lim }_{x\to c^{-}}f(x)\ne {\lim }_{x\to c^{+}}f(x)$, or if either one-sided limit fails to exist, then the two-sided limit ${\lim }_{x\to c}f(x)$does not exist (DNE).

Core Concepts & Rules

• Limit vs. Function Value: A limit describes the value a function approaches as $x$ nears a point $c$. This value is independent of the function's actual value at$c$, $f(c)$.

• Existence of a Limit: For a (two-sided) limit to exist at $x=c$, the function must approach the exact same finite value from the left side of $c$ as it does from the right side of $c$.

• One-Sided Limits: Limits can be evaluated from a single direction. The left-hand limit ($x\to c^{-}$) considers values of $x$ less than $c$, and the right-hand limit ($x\to c^{+}$) considers values of $x$ greater than $c$.

• The Biconditional Rule: The two-sided limit ${\lim }_{x\to c}f(x)=L$ if and only if ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)=L$. If the one-sided limits are not equal, the two-sided limit does not exist.

• Limit Notation: The expression ${\lim }_{x\to c}f(x)=L$ is the standard notation used to formally state the value of a limit.

Step-by-Step Example 1: Interpreting Limits from a Table

Problem: The table below gives values for a function $g(x)$ at selected $x$-values near $x=4$. Use the table to determine ${\lim }_{x\to 4^{-}}g(x)$, ${\lim }_{x\to 4^{+}}g(x)$, and ${\lim }_{x\to 4}g(x)$. Note that $g(4)$ is undefined.

Step 1: Determine the left-hand limit, ${\lim }_{x\to 4^{-}}g(x)$.

To find the left-hand limit, we examine the values of $g(x)$ as $x$ approaches 4 from the left side (i.e., for $x<4$).

• Look at the $x$-values in the table that are less than 4: 3.9, 3.99, and 3.999.

• Observe the corresponding $g(x)$-values: 6.81, 6.98, and 6.998.

• As $x$ gets closer to 4 from the left, $g(x)$ appears to be getting closer to 7.

• Therefore, we conclude: ${\lim }_{x\to 4^{-}}g(x)=7$.

Step 2: Determine the right-hand limit, ${\lim }_{x\to 4^{+}}g(x)$.

To find the right-hand limit, we examine the values of $g(x)$ as $x$ approaches 4 from the right side (i.e., for $x>4$).

• Look at the $x$-values in the table that are greater than 4: 4.1, 4.01, and 4.001.

• Observe the corresponding $g(x)$-values: 7.21, 7.02, and 7.002.

• As $x$ gets closer to 4 from the right, $g(x)$ appears to be getting closer to 7.

• Therefore, we conclude: ${\lim }_{x\to 4^{+}}g(x)=7$.

Step 3: Determine the two-sided limit, ${\lim }_{x\to 4}g(x)$.

The two-sided limit exists if and only if the left-hand and right-hand limits exist and are equal.

• From Step 1, ${\lim }_{x\to 4^{-}}g(x)=7$.

• From Step 2, ${\lim }_{x\to 4^{+}}g(x)=7$.

• Since both one-sided limits are equal to 7, the two-sided limit exists and is also equal to 7.

• Therefore, ${\lim }_{x\to 4}g(x)=7$. Notice that this is true even though $g(4)$ is undefined.

Step-by-Step Example 2: Interpreting Limits from a Graph

Problem: Use the graph of the function $f(x)$ below to find the following values:

a) ${\lim }_{x\to -2}f(x)$

b) ${\lim }_{x\to 3^{-}}f(x)$

c) ${\lim }_{x\to 3^{+}}f(x)$

d) ${\lim }_{x\to 3}f(x)$

e) $f(3)$