Estimating Limit Values from Tables

Samuel Baker

7 min read

Next Topic - Determining Limits Using Algebraic Properties of Limits

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Study Guide Overview

This study guide covers estimating limits from tables, focusing on the concept of a limit, particularly one-sided limits. It explains how to use tables to estimate limits when direct substitution fails, including choosing appropriate x-values and interpreting the resulting y-values. The guide provides practice questions and emphasizes the importance of checking if left-hand and right-hand limits are equal for the limit to exist.

#AP Calculus AB/BC: Estimating Limits from Tables 📈

Welcome to your ultimate guide for mastering limits using tables! This guide is designed to be your go-to resource, especially the night before the exam. Let's make sure you're confident and ready to ace it!

Key Concept

Remember, a limit is the value a function approaches as the input (x) gets closer to a specific value. It's all about what's happening near a point, not necessarily at the point.

#Concept of a Limit Recap

As we've discussed, a limit describes the behavior of a function as its input approaches a certain value. The notation looks like this:

$\lim_{x \to a} f(x) = L$

Here, 'x' approaches 'a', and 'L' is the limit. We're diving into how to find 'L' using tables. 🔢

#One-Sided Limits

Before we jump into tables, it's crucial to understand one-sided limits. These look at the function's behavior as 'x' approaches 'a' from either the left (values less than 'a') or the right (values greater than 'a').

Approaching from the right (values greater than 'a'): $\lim_{x \to a^+} f(x) = L$
Approaching from the left (values less than 'a'): $\lim_{x \to a^-} f(x) = L$

Think of it like approaching a destination from different directions. ⬅️➡️

Memory Aid

Remember: a+ means approach from the right (positive side), and a- means approach from the left (negative side).

#Using a Table to Estimate Limits

Sometimes, direct substitution doesn't work (we get an indeterminate form like $\frac{0}{0}$ ). In these cases, we use a table to see what happens as 'x' gets close to 'a'. ❌

For example:

$\lim_{x \to 0} \frac{x}{\sqrt{x+1}-1}$

Direct substitution gives us $\frac{0}{0}$ , which is undefined. So, we'll use a table! ↔️

Here's how a table helps:

Choose x-values close to 'a' from both sides.
Calculate the corresponding y-values (f(x)).
Observe if the y-values approach a common value from both sides. 📚

![Screenshot 2023-10-26 at 11.42.42 AM.png](Screenshot 2023-10-26 at 11.42.42 AM.png)

Image courtesy of Calculus: 9th Edition by Larson and Edward.

In the table above:

As x approaches 0 from the right, y approaches 2. - As x approaches 0 from the left, y also approaches 2. Therefore, we estimate that the limit of the function at x = 0 is 2. 📱

Quick Fact

When we say x-values are "near" or "close" to the target value 'a', we mean numbers in close proximity to 'a' from the left and right sides. For example, if a = 1, then 0.9, 0.99, 0.999 are close from the left, and 1.001, 1.01, 1.1 are close from the right. 💡

Remember, for this method to work, the left and right limits must approach the same value. If they don't, it might indicate a vertical asymptote. 📈

#Practice Evaluating Limits from a Table

Let's tackle another example (courtesy of Calculus 9th Edition by Larson and Edward):

$\lim_{x \to 4} \frac{x-4}{x^2-3x-4}$

First, we create a table with x-values near 4:

![Screenshot 2023-10-26 at 12.15.44 PM.png](Screenshot 2023-10-26 at 12.15.44 PM.png)

Image courtesy of Calculus 9th Edition by Larson and Edward.

Now, we calculate the corresponding y-values:

![Screenshot 2023-10-26 at 12.20.06 PM.png](Screenshot 2023-10-26 at 12.20.06 PM.png)

Image courtesy of Calculus 9th Edition by Larson and Edward

From the table, we can see that as x approaches 4 from both sides, the y-values approach 0.2. So, we estimate:

$\lim_{x \to 4} \frac{x-4}{x^2-3x-4} = 0.2$

Exam Tip

The key is to identify x-values close to the target and observe the trend of the y-values. Look for a common value that the y-values are approaching from both directions! 💾

#Conclusion

Using tables is a powerful way to estimate limits, especially when direct substitution fails. It's all about observing the behavior of the function as x gets closer and closer to the target value. You've got this! ✅

#Final Exam Focus

Here’s what to focus on for the exam:

High-Value Topics:
- Limits and their properties
- One-sided limits
- Estimating limits from tables
Common Question Types:
- Multiple-choice questions testing understanding of limits
- Free-response questions requiring you to estimate limits from a table

Exam Tip

Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.

Common Mistake

Common Pitfalls: Make sure to check if the limits from both sides are approaching the same value. If they are not, the limit does not exist.

Key Concept: Remember that the limit exists only if the left-hand and right-hand limits are equal. This is a frequently tested concept!

#Practice Questions

Practice Question

Multiple Choice Questions

Given the table below, what is $\lim_{x \to 2} f(x)$ ?

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	3.8	3.98	3.998	4.002	4.02	4.2

(A) 2 (B) 3 (C) 4 (D) Does not exist

Use the table below to estimate $\lim_{x \to 1} g(x)$ .

x	0.9	0.99	0.999	1.001	1.01	1.1
g(x)	2.8	2.98	2.998	3.002	3.02	3.2

(A) 1 (B) 2 (C) 3 (D) Does not exist

What is the limit of the function $h(x)$ as $x$ approaches 3, based on the table below?

x	2.9	2.99	2.999	3.001	3.01	3.1
h(x)	5.8	5.98	5.998	6.002	6.02	6.2

(A) 3 (B) 5 (C) 6 (D) Does not exist

Free Response Question

Consider the function $f(x) = \frac{x-1}{\sqrt{x}-1}$ .

(a) Create a table of values for $x$ near 1 (both less than and greater than 1). (b) Use your table to estimate $\lim_{x \to 1} f(x)$ . (c) Explain why direct substitution does not work in this case.

Scoring Rubric:

(a) Table of values (3 points): - 1 point for choosing appropriate values near 1. - 1 point for correct calculations of f(x) for values less than 1. - 1 point for correct calculations of f(x) for values greater than 1. (b) Estimation of the limit (2 points): - 1 point for stating the estimated limit. - 1 point for justification based on the table. (c) Explanation of why direct substitution fails (1 point): - 1 point for correctly explaining that direct substitution leads to an indeterminate form.

Answers:

Multiple Choice:

Free Response:

(a) Table of values:

x	0.9	0.99	0.999	1.001	1.01	1.1
f(x)	1.948	1.995	1.999	2.001	2.005	2.049

(b) $\lim_{x \to 1} f(x) = 2$ . The y-values approach 2 as x approaches 1 from both sides. (c) Direct substitution results in $\frac{0}{0}$ , which is an indeterminate form. Therefore, we must use another method to determine the limit.