Estimating Limit Values from Tables - AP Calculus AB Study Guide

The Core Idea: Estimating Limit Values from Tables

The concept of a limit describes the value that a function "approaches" as the input variable gets closer and closer to a specific number. When we analyze a function using a table of values, we are not concerned with the actual value of the function at the specific point, which may or may not be defined. Instead, we are investigating the trend of the output values (the $y$-values) as we select input values (the $x$-values) that are infinitesimally close to our target number from both the left side (numbers slightly smaller) and the right side (numbers slightly larger).

This process of examining the function's behavior from both sides is crucial. The overall, or two-sided, limit exists only if the function approaches the exact same value from the left as it does from the right. If the function trends toward different values from each side, we conclude that the limit does not exist. A table provides the discrete data points needed to observe these trends and make an educated estimation of the limit.

Key Definitions and Concepts

The estimation of limits from a table relies on the foundational concepts of one-sided and two-sided limits.

• Right-Hand Limit: The right-hand limit, denoted ${\lim }_{x\to c^{+}}f(x)$, is the value that $f(x)$ approaches as $x$ gets closer and closer to $c$ from values greater than $c$.

• Left-Hand Limit: The left-hand limit, denoted ${\lim }_{x\to c^{-}}f(x)$, is the value that $f(x)$ approaches as $x$ gets closer and closer to $c$ from values less than $c$.

• The (Two-Sided) Limit: The overall limit, denoted ${\lim }_{x\to c}f(x)$, exists if and only if the left-hand limit and the right-hand limit are equal.

  • Condition for Existence: If ${\lim }_{x\to c^{-}}f(x)=L$ and ${\lim }_{x\to c^{+}}f(x)=L$, then ${\lim }_{x\to c}f(x)=L$.

  • Condition for Non-Existence: If ${\lim }_{x\to c^{-}}f(x)\ne {\lim }_{x\to c^{+}}f(x)$, then ${\lim }_{x\to c}f(x)$ does not exist.

Understanding the Limit vs. the Function Value

A critical distinction in calculus is that the limit of a function as $x$ approaches $c$ is not necessarily the same as the value of the function at$c$, which is $f(c)$. The limit is concerned with the intended height of the function at $c$ based on the path it takes from both sides. The actual value $f(c)$ is the actual height of the function at that exact point.

There are three common scenarios you will encounter when using a table:

1. The limit as $x\to c$ exists, and it is equal to $f(c)$.

2. The limit as $x\to c$ exists, but it is not equal to $f(c)$, or $f(c)$ is undefined.

3. The limit as $x\to c$ does not exist, regardless of the value of $f(c)$.

A table of values helps us focus only on the values near$c$, which is precisely what is required to estimate the limit. The single entry for $x=c$ in the table, if it exists, is irrelevant to the process of finding the limit itself.

Core Concepts & Rules

• A limit is estimated by observing the trend of output values ($f(x)$) in a table as the input values ($x$) get progressively closer to a target number, $c$.

• To properly estimate a limit, one must examine the function's behavior from both sides of $c$: the left side (inputs less than $c$) and the right side (inputs greater than $c$).

• The two-sided limit, ${\lim }_{x\to c}f(x)$, exists and is equal to $L$ if and only if the function values approach $L$ from both the left and the right.

• If the function values approach one number from the left and a different number from the right, the two-sided limit does not exist.

Step-by-Step Example 1: Estimating a Limit That Exists

Problem: The function $f$ is defined for all real numbers except $x=3$. The table below gives values of $f(x)$ for selected values of $x$. Estimate the value of ${\lim }_{x\to 3}f(x)$.

Step 1: Analyze the Left-Hand Limit

Look at the $x$-values that are approaching 3 from the left (values less than 3): 2.9, 2.99, and 2.999.

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

As $x$ gets closer to 3 from the left, the values of $f(x)$ appear to be getting closer to 7.

So, we estimate that ${\lim }_{x\to 3^{-}}f(x)=7$.

Step 2: Analyze the Right-Hand Limit

Look at the $x$-values that are approaching 3 from the right (values greater than 3): 3.1, 3.01, and 3.001.

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

As $x$ gets closer to 3 from the right, the values of $f(x)$ also appear to be getting closer to 7.

So, we estimate that ${\lim }_{x\to 3^{+}}f(x)=7$.

Step 3: Compare the One-Sided Limits

The left-hand limit is 7 and the right-hand limit is 7. Since they are equal, the two-sided limit exists and is equal to this value.

Conclusion:

Based on the table, we estimate that ${\lim }_{x\to 3}f(x)=7$. Note that the fact that $f(3)$ is undefined is irrelevant to the value of the limit.

Step-by-Step Example 2: A Limit That Does Not Exist

Problem: The function $g$ is continuous at all real numbers except $x=1$. The table below gives values of $g(x)$ for selected values of $x$. What is the value of ${\lim }_{x\to 1}g(x)$?

Step 1: Analyze the Left-Hand Limit

Look at the $x$-values approaching 1 from the left: 0.9, 0.99, 0.999.

Observe the corresponding $g(x)$-values: 4.7, 4.97, 4.997.

The trend shows that as $x$ approaches 1 from the left, $g(x)$ is approaching 5.

We estimate that ${\lim }_{x\to 1^{-}}g(x)=5$.

Step 2: Analyze the Right-Hand Limit

Look at the $x$-values approaching 1 from the right: 1.1, 1.01, 1.001.

Observe the corresponding $g(x)$-values: 2.3, 2.03, 2.003.

The trend shows that as $x$ approaches 1 from the right, $g(x)$ is approaching 2.

We estimate that ${\lim }_{x\to 1^{+}}g(x)=2$.

Step 3: Compare the One-Sided Limits

The left-hand limit is 5, and the right-hand limit is 2. Since ${\lim }_{x\to 1^{-}}g(x)\ne {\lim }_{x\to 1^{+}}g(x)$, the two-sided limit does not exist.

Conclusion:

${\lim }_{x\to 1}g(x)$ does not exist. The fact that $g(1)=8$ has no bearing on whether the limit exists.

Using Your Calculator

While AP Exam questions will provide you with a table, you can use a graphing calculator to generate your own table to estimate a limit if you are given a function's equation.

Goal: Estimate ${\lim }_{x\to 2}\frac{x^2-4}{x-2}$.

Method 1: Using the Automatic Table

1. Press Y= and enter the function into Y1: Y1=(X^2-4)/(X-2).

2. Press 2nd then WINDOW to access TBLSET (Table Setup).

3. Set TblStart to the value $x$ is approaching. Here, TblStart = 2.

4. Set ΔTbl (Delta Table) to a very small number, like $0.001$. This sets the increment for the $x$-values in the table.

5. Press 2nd then GRAPH to access the TABLE.

6. You will see an ERROR at $x=2$ because the function is undefined there. However, by scrolling up and down, you can observe the $y$-values for $x$-values just below and just above 2. You will see values like 3.999 for $x=1.999$ and 4.001 for $x=2.001$, suggesting the limit is 4.

Method 2: Using the "Ask" Feature

1. Enter the function into Y1 as before.

2. Go to TBLSET.

3. Change the Indpnt: setting from Auto to Ask.

4. Go to the TABLE. It will be blank.

5. Manually enter x-values that get closer and closer to 2 from both sides (e.g., 1.9, 1.99, 1.999, 2.001, 2.01) and press ENTER after each one. The calculator will compute the corresponding Y1 value, allowing you to observe the trend.

AP Exam Quick Hit

Common Question Types

• Estimating a Two-Sided Limit: You will be given a table of values for a function $f(x)$ around a point $c$ and asked to find ${\lim }_{x\to c}f(x)$. You must check both sides and determine if they approach the same value.

  • Example: Given a table for $f(x)$ around $x=5$, estimate ${\lim }_{x\to 5}f(x)$. The answer could be a specific number or "Does Not Exist".

• Estimating a One-Sided Limit: You will be given a table and asked specifically for the left-hand limit (${\lim }_{x\to c^{-}}f(x)$) or the right-hand limit (${\lim }_{x\to c^{+}}f(x)$). In this case, you only need to analyze the trend from the specified side.

  • Example: Given the same table for $f(x)$ around $x=5$, estimate ${\lim }_{x\to 5^{+}}f(x)$.

Common Mistakes

• Confusing the Limit with the Function Value: A common mistake is to state that the limit is the value of the function at the point. For instance, in Example 2 above, a student might incorrectly state the limit is 8 because $g(1)=8$. The limit is about the trend near the point, not the value at the point.

• Only Checking One Side: When asked for a two-sided limit (${\lim }_{x\to c}f(x)$), students sometimes only look at the values from the left or the right and assume that is the limit. You must always check both sides for a two-sided limit.

• Assuming Undefined Means DNE: If the table shows "Undefined" or "Error" for $f(c)$, students may jump to the conclusion that the limit "Does Not Exist". The limit can exist even if the function is undefined at that point (as shown in Example 1).

• Averaging the Left and Right Limits: If the left-hand limit is 5 and the right-hand limit is 2, some students might incorrectly average them to get 3.5. If the one-sided limits are not equal, the two-sided limit simply does not exist.