Free for students · Ad-free · WCAG 2.1 AA Compliant · Accessibility

Estimating Limit Values from Tables

Lesson ~10 min read

In simple terms: In simple terms, this topic is about using a list of input and output values to guess what a function's output is trying to reach at a specific input, even if it never quite gets there.

Why this matters

Imagine you're tracking a friend, Marcus, who is running a marathon in Chicago. You don't have his exact live location, but his GPS watch sends you an update every few minutes. You see his location at mile marker 12.9, then 12.95, then 12.99. From the other side, you see updates from 13.1, then 13.05, then 13.01.

You don't have a data point for exactly mile 13, but by looking at the data from both sides, you can make a very strong prediction about where he was at that moment.

Calculus uses this same idea. We can "zoom in" on a function around a specific point using a table of values. By watching where the output values are heading as the input gets closer and closer from both sides, we can estimate the function's limit. Today, we'll master this detective work.

Concept overview

flowchart TD
    A[Start: Estimate limit of f(x) as x approaches c] --> B{Examine f(x) for x-values getting closer to c};
    B --> C[From the Left (x < c)];
    B --> D[From the Right (x > c)];
    C --> E{What value is f(x) approaching? Let's call it L.};
    D --> F{What value is f(x) approaching? Let's call it R.};
    subgraph "Compare the Approaches"
        G{Is L = R?};
    end
    E --> G;
    F --> G;
    G -- Yes --> H[The Limit is L];
    G -- No --> I[The Limit Does Not Exist];
    J[Note: The value of f(c) itself does not affect the limit!];

Core explanation

Hello! I'm Saavi, and I'm so glad you're here. Today, we're diving into one of the foundational skills of calculus: estimating limits using a table. It feels a bit like detective work, and once you get the hang of it, it's incredibly satisfying.

What is a Limit, Really?

Before we look at tables, let's refresh the big idea. A limit is the value that a function approaches as the input gets closer and closer to some number.

Think of it like this: You're walking towards a specific spot on a basketball court—the free-throw line. The limit is the exact location of that line. It doesn't matter if you're walking from half-court or from under the basket. As you get infinitely close, you are approaching that one specific spot. The limit is the destination, not the path you take to get there.

In this lesson, we don't have a graph or an equation for our function. All we have is a table of x and f(x) values, like a log of data points. Our job is to use these clues to figure out what value f(x) is approaching.

The Two-Sided Approach: The Golden Rule

To estimate a limit as x approaches a value c, you must look at the function's behavior from both sides:

  1. 1
    From the left
    Using x-values that are slightly less than c.
  2. 2
    From the right
    Using x-values that are slightly greater than c.

The limit exists only if the function approaches the same single value from both the left and the right.

Let's see this in action. Suppose we want to estimate lim x→2 f(x) using the following table:

x 1.9 1.99 1.999 2.0 2.001 2.01 2.1
f(x) 3.85 3.985 3.998 ??? 4.002 4.015 4.15
  1. 1
    Approach from the left
    Look at the x-values getting closer to 2 from below: 1.9, 1.99, 1.999. What are the corresponding f(x) values doing? They are 3.85, 3.985, 3.998. It looks like they are getting closer and closer to 4.
  2. 2
    Approach from the right
    Now look at the x-values getting closer to 2 from above: 2.1, 2.01, 2.001. The f(x) values are 4.15, 4.015, 4.002. These also seem to be zeroing in on 4.

Since the function approaches 4 from the left AND from the right, we can confidently estimate: lim x→2 f(x) = 4

The Most Important (and Often Confused) Point

Notice the "???" in the table at x = 2. For estimating the limit, the actual value of f(2) is completely irrelevant.

  • f(2) could be 4.
  • f(2) could be 100.
  • f(2) could be undefined (a "hole" in the function).

When Does a Limit Not Exist?

So, what does it look like in a table when a limit doesn't exist (we often write this as DNE)? This happens when the left and right approaches don't agree.

Consider this table for a function g(x) as we try to find lim x→1 g(x):

x 0.9 0.99 0.999 1.0 1.001 1.01 1.1
g(x) 1.9 1.99 1.999 5.0 3.001 3.01 3.1
  • From the left
    As x approaches 1, the g(x) values (1.9, 1.99, 1.999) are clearly approaching 2.
  • From the right
    As x approaches 1, the g(x) values (3.1, 3.01, 3.001) are clearly approaching 3.

Since the approach from the left (2) does not equal the approach from the right (3), the overall two-sided limit does not exist.

It doesn't matter that g(1) = 5. That's just extra information designed to test if you understand the concept. The two paths don't lead to the same meeting point, so there's no single limit. Think of it as two people on different floors of a building walking towards the same spot on a map—they'll never meet.

That's the core idea. You are a data detective. You look at the clues from the left, the clues from the right, and see if they point to the same conclusion.

Worked examples

Let's walk through a few examples together. The key is to be systematic: check the left, check the right, then compare.

Example 1

A Straightforward Limit

Problem: Estimate lim x→-3 h(x) using the table below.

x -3.1 -3.01 -3.001 -3.0 -2.999 -2.99 -2.9
h(x) 8.8 8.99 8.999 Undef. 9.001 9.01 9.2

Solution Walkthrough:

  1. 1
    Identify the Goal
    We need to find the limit as x gets incredibly close to -3.
  2. 2
    Check the Left Approach
    We look at x values less than -3 that are getting closer to it: -3.1, -3.01, -3.001.
    • The corresponding h(x) values are 8.8, 8.99, and 8.999.
    • Why this step matters: We are establishing the trend from one side. This pattern suggests the function is approaching 9 from the left.
  3. 3
    Check the Right Approach
    Now, we look at x values greater than -3 that are getting closer: -2.9, -2.99, -2.999.
    • The corresponding h(x) values are 9.2, 9.01, and 9.001.
    • Why this step matters: We need to see if the trend from the other side confirms our initial finding. This pattern also suggests the function is approaching 9.
  4. 4
    Compare and Conclude
    The approach from the left is 9, and the approach from the right is 9. They match!
    • Notice that h(-3) is undefined. As we've discussed, this doesn't affect the limit.
    • Final Answer: Because both sides agree, we can estimate that lim x→-3 h(x) = 9.
Example 2

A Limit That Does Not Exist

Problem: Using the table for the function k(x), estimate lim x→4 k(x).

x 3.9 3.99 3.999 4.0 4.001 4.01 4.1
k(x) 10.5 10.1 10.01 10 -0.01 -0.1 -0.5

Solution Walkthrough:

  1. 1
    Identify the Goal
    We're investigating the behavior of k(x) around x = 4.
  2. 2
    Check the Left Approach
    Look at x values 3.9, 3.99, 3.999.
    • The k(x) values are 10.5, 10.1, 10.01.
    • This trend is clearly heading towards 10.
  3. 3
    Check the Right Approach
    Look at x values 4.1, 4.01, 4.001.
    • The k(x) values are -0.5, -0.1, -0.01.
    • This trend is heading towards 0.
  4. 4
    Compare and Conclude
    The left-hand approach is 10, but the right-hand approach is 0.
    • Why this is critical: Since 10 ≠ 0, the function is "jumping" at x = 4. The two sides do not meet.
    • Even though the function is defined at x=4 (we see k(4)=10), this doesn't create a limit. The right-hand side's disagreement is what matters.
    • Final Answer: lim x→4 k(x) Does Not Exist (DNE).

Try it yourself

Time to be the detective. Use what you've learned to solve these.

Problem 1

The following table shows values for a function p(x). Estimate lim x→5 p(x).

x 4.9 4.99 4.999 5.0 5.001 5.01 5.1
p(x) -2.5 -2.95 -2.995 7 -3.005 -3.05 -3.5

Problem 2

A function q(x) is defined by the table below. Estimate lim x→-1 q(x).

x -1.1 -1.01 -1.001 -1.0 -0.999 -0.99 -0.9
q(x) 6.1 6.01 6.001 Undef. 3.999 3.99 3.9
Back to AP Calculus BC Take a mock exam