Estimating Limit Values from Tables

Lesson ~10 min read

In simple terms: In simple terms, estimating limits from tables is about using a list of a function's input and output values to predict what output the function is heading towards as the input gets closer and closer to a specific number.

Why this matters

Imagine you're tracking a package on your phone. The GPS updates its location every few seconds, showing it getting closer and closer to your home in Chicago. You see it at 10:00 AM on Jefferson St, then at 10:01 AM on Adams St, then at 10:02 AM on Monroe St. Based on this data, you can predict with high confidence that its destination is your apartment building on Madison St, even if you can't see the delivery truck at that exact final moment.

Estimating limits from a table is a lot like that. We're given a set of data points—the GPS updates—and we use them to predict the function's ultimate destination, or limit. We'll learn how to read these tables, look at the clues from both sides, and make a confident prediction about the limit, even if the function has a "connection lost" error at the final destination.

Concept overview

flowchart TD
    A[Start: Given a table and target x=c] --> B{Look at x-values < c};
    B --> C[What y-value is approached? --> L];
    A --> D{Look at x-values > c};
    D --> E[What y-value is approached? --> R];
    C --> F{Is L = R?};
    E --> F;
    F -- Yes --> G[Limit is L];
    F -- No --> H[Limit Does Not Exist (DNE)];
    B --> I{Are y-values growing without bound?};
    I -- Yes --> J[Limit is ∞ or -∞];
    D --> I;

Core explanation

Hello! I'm Saavi, and I'm excited to walk you through one of the foundational ideas in calculus: limits. Today, we're going to be data detectives and learn how to estimate a limit just by looking at a table of values.

What Are We Even Doing?

At its heart, a limit in calculus is about describing what happens to a function's output (f(x)) as its input (x) gets infinitely close to a particular number. The key word here is approach. We care about the journey, not necessarily the arrival.

Think of it like two friends, Maya and Carlos, walking along a path on a hilly park trail. They agree to meet at the spot where x = 3 miles. We want to know the altitude they are approaching.

A table of values is like a log of their positions:

x (miles)	f(x) (altitude in feet)
2.9	48.7
2.99	49.97
2.999	49.997
3.0	??? (hole in the path!)
3.001	50.003
3.01	50.03
3.1	50.3

Approaching from Two Sides

To find the limit, we must check the approach from both sides.

1. The Left-Hand Approach: Look at Maya's journey. She is approaching x = 3 from the left side, with x-values that are slightly less than 3.

At x = 2.9, her altitude is 48.7 ft.
At x = 2.99, her altitude is 49.97 ft.
At x = 2.999, her altitude is 49.997 ft.

What number do these altitude values seem to be zeroing in on? It looks like they are getting incredibly close to 50. We say the left-hand limit is 50.

2. The Right-Hand Approach: Now look at Carlos's journey. He is approaching x = 3 from the right side, with x-values that are slightly greater than 3.

At x = 3.1, his altitude is 50.3 ft.
At x = 3.01, his altitude is 50.03 ft.
At x = 3.001, his altitude is 50.003 ft.

What number are his altitude values approaching? It also looks like they are getting incredibly close to 50. We say the right-hand limit is 50.

The Big Conclusion: Does the Limit Exist?

The overall limit exists if and only if the left-hand limit equals the right-hand limit.

Since Maya (from the left) and Carlos (from the right) are both approaching an altitude of 50 feet, we can confidently say:

The limit of f(x) as x approaches 3 is 50.

When the Limit Does Not Exist (DNE)

What if the table looked like this?

x (miles)	g(x) (altitude in feet)
1.9	3.85
1.99	3.99
2.0	10
2.01	7.02
2.1	7.2

Let's find the limit of g(x) as x approaches 2.

From the left
As x approaches 2 from values like 1.9 and 1.99, the g(x) values (3.85, 3.99) are clearly approaching 4.
From the right
As x approaches 2 from values like 2.1 and 2.01, the g(x) values (7.2, 7.02) are clearly approaching 7.

Since the left-hand limit (4) does not equal the right-hand limit (7), the overall limit Does Not Exist (we write DNE).

This is like Maya approaching an altitude of 4 feet, while Carlos approaches an altitude of 7 feet. They aren't heading to the same spot! This is a "jump" in the function. And notice, the actual value g(2) = 10 is completely irrelevant to finding the limit. It's a classic distracter.

Unbounded Behavior

Sometimes, the function doesn't approach a specific number, but instead shoots up to infinity or down to negative infinity.

Consider this table for the limit as x approaches 0:

x	h(x)
-0.1	100
-0.01	10,000
-0.001	1,000,000
0	undefined
0.001	1,000,000
0.01	10,000
0.1	100

From both the left and the right, as x gets closer to 0, the h(x) values are getting huge without any ceiling. They are growing without bound. In this case, we say the limit is infinity. Technically, the limit "does not exist" because infinity isn't a real number, but saying the limit is ∞ is more descriptive of the function's behavior. Your teacher will clarify how they want you to write this, but for the AP Exam, DNE is often the safe choice unless asked for more specific behavior.

The process is always the same: check the left, check the right, and compare. You've got this.

Worked examples

Let's work through a few examples together, just like we would on an AP Exam free-response question.

Example 1

The Limit Exists

Problem: The function f is continuous for all real numbers except x = 4. The table below gives values of f at selected x values. What is the best estimate for the limit of f(x) as x approaches 4?

x	3.9	3.99	3.999	4	4.001	4.01	4.1
`f(x)`	8.82	8.98	8.998	undef	9.002	9.02	9.22

Solution Walkthrough:

1
Identify the Goal
We need to find the limit as x approaches 4. The notation for this is lim (x→4) f(x).
2
Check the Left-Hand Approach
Look at the x values that are less than 4 and getting closer to it: 3.9, 3.99, 3.999.
- The corresponding f(x) values are 8.82, 8.98, and 8.998.
- Ask yourself: "What number are these values creeping up on?" They seem to be getting extremely close to 9. So, the left-hand limit is 9.
3
Check the Right-Hand Approach
Look at the x values that are greater than 4 and getting closer to it: 4.1, 4.01, 4.001.
- The corresponding f(x) values are 9.22, 9.02, and 9.002.
- Ask yourself: "What number are these values settling down on?" They also seem to be getting extremely close to 9. So, the right-hand limit is 9.
4
Compare and Conclude
The left-hand limit (9) is equal to the right-hand limit (9).
- Therefore, the limit exists and is equal to 9.

Example 2

The Limit Does Not Exist (DNE)

Problem: The table below gives values of a function g. What is the best estimate for the limit of g(x) as x approaches -2?

x	-2.1	-2.01	-2.001	-2	-1.999	-1.99	-1.9
`g(x)`	-5.1	-5.01	-5.001	10	4.999	4.99	4.9

Solution Walkthrough:

1
Identify the Goal
We need to find lim (x→-2) g(x).
2
Check the Left-Hand Approach
Look at x values less than -2: -2.1, -2.01, -2.001.
- The corresponding g(x) values are -5.1, -5.01, -5.001.
- These values are clearly approaching -5. The left-hand limit is -5.
3
Check the Right-Hand Approach
Look at x values greater than -2: -1.9, -1.99, -1.999.
- The corresponding g(x) values are 4.9, 4.99, 4.999.
- These values are clearly approaching 5. The right-hand limit is 5.
4
Compare and Conclude
The left-hand limit (-5) is not equal to the right-hand limit (5).
- Therefore, the limit as x approaches -2 Does Not Exist (DNE).

Why it works this way: The function "jumps" at x = -2. From one side, it's heading towards a height of -5, and from the other, it's heading towards a height of 5. Since there's no single, agreed-upon height, there's no limit. Notice again that the actual value g(-2) = 10 is a red herring—it has no bearing on the limit calculation.

Try it yourself

Ready to try a couple on your own? Take your time, and remember the process.

Problem 1: The function h(x) is given by the table below. What is the best estimate for the limit of h(x) as x approaches 1?

x	0.9	0.99	0.999	1	1.001	1.01	1.1
`h(x)`	-1.718	-1.730	-1.731	-1.732	-1.733	-1.735	-1.748

Problem 2: The cost C(w) in dollars to ship a package weighing w pounds from Dallas is given in the table. What is your estimate for the limit of C(w) as w approaches 3?

w (lbs)	2.8	2.9	2.99	3.0	3.01	3.1	3.2
`C(w)`	$12.50	$12.50	$12.50	$15.00	$15.00	$15.00	$15.00

TL;DR

In simple terms, estimating limits from tables is about using a list of a function's input and output values to predict what output the function is heading towards as the input gets closer and closer to a specific number.

Syntax pattern

The limit of f(x) as x approaches 3 is 50.

Key terms

AP Calculus Limits from tables Numerical limits Estimating limits Limit does not exist One-sided limits AP Calculus AB Unit 1 LIM-1.C Shrutam ===END===

You can now…

LIM-1.C: Estimate limits of functions.

Essential knowledge (exam-tested)

LIM-1.C.5: Numerical information can be used to estimate limits.

Concept map

flowchart TD
    A[Start: Given a table and target x=c] --> B{Look at x-values < c};
    B --> C[What y-value is approached? --> L];
    A --> D{Look at x-values > c};
    D --> E[What y-value is approached? --> R];
    C --> F{Is L = R?};
    E --> F;
    F -- Yes --> G[Limit is L];
    F -- No --> H[Limit Does Not Exist (DNE)];
    B --> I{Are y-values growing without bound?};
    I -- Yes --> J[Limit is ∞ or -∞];
    D --> I;

Read what Saavi narrates

Hi there, I'm Saavi. Welcome to Shrutam.

Let's talk about a cool concept in calculus called limits.

Imagine you're tracking a package on your phone. The GPS updates its location every few seconds, showing it getting closer to your home. Based on that data... that list of locations and times... you can predict exactly where it's going, even if you can't see the delivery truck at the final moment.

Estimating limits from a table is just like that. We're given a set of data points, and we use them to predict the function's ultimate destination. The main idea is to look at a table of x and y values. By observing what y-value the function gets closer to as x approaches a specific number from both the left side and the right side, we can estimate the function's limit.

Let's try one. Imagine a table for a function, let's call it f of x. We want to find the limit as x approaches 4. The table shows us that when x is 3.9, f of x is 8.82. When x is 3.99, f of x is 8.98. As we get even closer, at 3.999, the function value is 8.998. What number does that seem to be approaching? It looks like it's getting really, really close to 9.

Now let's check from the other side. The table shows that when x is 4.1, f of x is 9.22. When x is 4.01, f of x is 9.02. And at 4.001, the function value is 9.002. Again, what number is that trend pointing to? It's also pointing to 9.

Since the approach from the left and the approach from the right both lead to the same number, 9, we can say the limit is 9.

Now, here is the most common mistake I see. In that table, the value of the function right at x equals 4 might be listed as "undefined." Many people see that and immediately say the limit doesn't exist. But that's not right! The limit is about the *approach*, not the destination itself. We only care that both sides were heading to the same place.

So remember, when you have a table, check the left, check the right, and if they match, that's your limit. You're just being a data detective. You can do this.

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