Estimating Limits
from Tables
Learn to determine the value a function approaches by examining numerical evidence from carefully constructed tables.
What Is a Limit?
A limit describes the value that \( f(x) \) approaches as \( x \) gets arbitrarily close to some number \( a \). We write:
Crucially, the limit cares about what happens near \( a \), not necessarily at \( a \). The function doesn't even need to be defined at \( x = a \) for the limit to exist.
Why Use Tables?
One-Sided Limits
To estimate \( \lim_{x \to a} f(x) \) from a table, we approach \( a \) from both directions:
Approaching from Both Sides
The Table Strategy
To estimate \( \lim_{x \to a} f(x) \) using a table, pick \( x \)-values that get progressively closer to \( a \) from each side:
If the \( f(x) \) values stabilize toward the same number from both sides, that number is your estimate for \( L \).
Limit vs. Function Value
How to Apply the Method
Example 1: \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \)
| \( x \) (from left) | \( f(x) \) | \( x \) (from right) | \( f(x) \) | |
|---|---|---|---|---|
| 1.9 | 3.9 | 2.1 | 4.1 | |
| 1.99 | 3.99 | 2.01 | 4.01 | |
| 1.999 | 3.999 | 2.001 | 4.001 | |
| 1.9999 | 3.9999 | 2.0001 | 4.0001 |
Example 1: Visual Confirmation
Example 2: \( \lim_{x \to 0} \frac{\sin x}{x} \)
| \( x \) (from left) | \( \frac{\sin x}{x} \) | \( x \) (from right) | \( \frac{\sin x}{x} \) | |
|---|---|---|---|---|
| \(-0.1\) | 0.99833 | 0.1 | 0.99833 | |
| \(-0.01\) | 0.99998 | 0.01 | 0.99998 | |
| \(-0.001\) | 0.9999998 | 0.001 | 0.9999998 |
Example 3: A Limit That Does Not Exist
Consider \( f(x) = \frac{|x|}{x} \). Let's estimate \( \lim_{x \to 0} f(x) \):
| \( x \) (from left) | \( \frac{|x|}{x} \) | \( x \) (from right) | \( \frac{|x|}{x} \) | |
|---|---|---|---|---|
| \(-0.1\) | \(-1\) | 0.1 | 1 | |
| \(-0.01\) | \(-1\) | 0.01 | 1 | |
| \(-0.001\) | \(-1\) | 0.001 | 1 |
Example 3: Jump Discontinuity
A jump discontinuity occurs when the one-sided limits exist but are not equal. The table immediately reveals this: all left values converge to one number and all right values converge to a different one.
Common Mistakes
Summary: Reading the Table
| Limit Exists | Limit DNE | |
|---|---|---|
| Left-hand values | Converge to \( L \) | Converge to \( L_1 \) |
| Right-hand values | Converge to \( L \) | Converge to \( L_2 \neq L_1 \) |
| Conclusion | \( \lim = L \) | DNE |
| \( f(a) \) matters? | No | No |
Question 1 of 5
A table shows these values for \( f(x) \) near \( x = 3 \):
From the left: \( f(2.9) = 6.8 \), \( f(2.99) = 6.98 \), \( f(2.999) = 6.998 \)
From the right: \( f(3.001) = 7.002 \), \( f(3.01) = 7.02 \), \( f(3.1) = 7.2 \)
What is the best estimate for \( \lim_{x \to 3} f(x) \)?
Question 2 of 5
A table for \( g(x) \) near \( x = 1 \) shows:
From left: \( g(0.9) = 4.1 \), \( g(0.99) = 4.01 \), \( g(0.999) = 4.001 \)
From right: \( g(1.001) = -2.001 \), \( g(1.01) = -2.01 \), \( g(1.1) = -2.1 \)
What is \( \lim_{x \to 1} g(x) \)?
Question 3 of 5
Given that \( \lim_{x \to 5} f(x) = 10 \) but \( f(5) = 3 \), which statement is true?
Question 4 of 5
To estimate \( \lim_{x \to 0} \frac{e^x - 1}{x} \), you build a table. Which set of \( x \)-values is most appropriate?
Question 5 of 5
A table for \( h(x) \) near \( x = 2 \) shows:
\( h(1.9) = 5.01 \), \( h(1.99) = 5.0001 \), \( h(1.999) = 5.000001 \)
\( h(2.001) = 4.999999 \), \( h(2.01) = 4.9999 \), \( h(2.1) = 4.99 \)
Also, \( h(2) = 5 \). What is \( \lim_{x \to 2} h(x) \)?