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Estimating Limits
from Tables

Learn to determine the value a function approaches by analyzing numerical data from both sides of a target point.

AP Calculus AB

What Is a Limit?

A limit describes the value that \( f(x) \) gets closer and closer to as \( x \) approaches a specific number \( a \). We write this as:

The key insight: we care about what happens near \( x = a \), not necessarily at \( x = a \). The function doesn't even need to be defined at \( a \) for the limit to exist.

Why Use Tables to Estimate Limits?

★ Numerical Evidence
Tables let you see concrete values of \( f(x) \) as \( x \) gets closer to the target.
★ No Formula Needed
Even when you can't simplify \( f(x) \) algebraically, a table reveals the trend.
★ Two-Sided Check
Approaching from left and right confirms whether the limit exists.

Left-Hand and Right-Hand Limits

To determine whether a limit exists at \( x = a \), we examine the function from both directions:

Left-Hand Limit
Values of \( x \) approaching \( a \) from below: \( x \to a^{-} \). We write \( \lim_{x \to a^{-}} f(x) \).
Right-Hand Limit
Values of \( x \) approaching \( a \) from above: \( x \to a^{+} \). We write \( \lim_{x \to a^{+}} f(x) \).

The Existence Condition

x y a L
\( x \to a^{-} \)
\( x \to a^{+} \)
Both \( \to L \)

When both sides converge to the same value \( L \), the two-sided limit equals \( L \): even if \( f(a) \) is undefined or different.

Limit Value vs. Function Value

\( \lim_{x \to a} f(x) \) \( f(a) \)
Meaning What \( f(x) \) approaches What \( f(x) \) equals at \( a \)
Requires \( f(a) \) defined? No Yes
Can differ from each other? Yes: a function can have \( f(a) \neq L \)
In a table Read the trend of nearby values Read the row where \( x = a \) exactly

Step-by-Step Strategy

1
Identify the target. Determine the value \( a \) that \( x \) is approaching.
2
Read left-side values. Look at \( f(x) \) for \( x \)-values slightly less than \( a \) (e.g., \( a - 0.1, a - 0.01, a - 0.001 \)).
3
Read right-side values. Look at \( f(x) \) for \( x \)-values slightly greater than \( a \).
4
Compare. If both sides approach the same number \( L \), then \( \lim_{x \to a} f(x) = L \).
5
If they differ, the two-sided limit does not exist (DNE).

Example 1: Estimating \( \lim_{x \to 2} f(x) \)

Consider \( f(x) = \dfrac{x^2 - 4}{x - 2} \). This is undefined at \( x = 2 \), but the limit may still exist. Let's build a table:

\( x \) (left) \( f(x) \) \( x \) (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: Visualized

x y 1 2 3 2 4 6 0
\( L = 4 \)
 from left from right

Example 2: When the Limit Does Not Exist

Estimate \( \lim_{x \to 0} \dfrac{|x|}{x} \) using a table:

\( x \) (left) \( f(x) \) \( x \) (right) \( f(x) \)
\(-0.1\) \(-1\) 0.1 1
\(-0.01\) \(-1\) 0.01 1
\(-0.001\) \(-1\) 0.001 1

Example 3: Famous Limit \( \lim_{x \to 0} \dfrac{\sin x}{x} \)

\( x \) (left) \( \frac{\sin x}{x} \) \( x \) (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

When Values Grow Without Bound

Estimate \( \lim_{x \to 0} \dfrac{1}{x^2} \) from a table:

\( x \) (left) \( \frac{1}{x^2} \) \( x \) (right) \( \frac{1}{x^2} \)
\(-0.1\) 100 0.1 100
\(-0.01\) 10,000 0.01 10,000
\(-0.001\) 1,000,000 0.001 1,000,000

Choosing Good \( x \)-Values

The quality of your estimate depends on how close your \( x \)-values get to the target. Use a pattern of powers of 10:

1
Start coarse: \( a \pm 0.1 \) to see the general trend.
2
Get closer: \( a \pm 0.01 \) and \( a \pm 0.001 \) for more decimal agreement.
3
Look for stabilization: When values agree to several decimal places, that's your estimate.

A few readings leave the intervals unseen

A probe touches a plain metal object and connects to a blank logger beside a notebook and separated blank reading cards.Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.

A few readings leave the intervals unseen

Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.

One-Sided Limits in Piecewise Functions

Some AP questions give a table for a piecewise function. The left and right sides may follow different rules:

\( x \) \( f(x) \) \( x \) \( f(x) \)
2.9 5.8 3.1 7.21
2.99 5.98 3.01 7.0201
2.999 5.998 3.001 7.002

Common Mistakes to Avoid

✕ Plugging In \( x = a \)
The limit is about what \( f(x) \) approaches, not \( f(a) \) itself. If \( f(a) \) is undefined, that's fine: the limit can still exist.
✕ Only Checking One Side
You must check both left and right. If they disagree, the two-sided limit DNE, even if one side has a clear value.
✕ Not Getting Close Enough
Values at \( a \pm 1 \) are too far away. Use values like \( a \pm 0.001 \) to get a reliable estimate.

Summary of Outcomes

Table Pattern Left-Hand Right-Hand Conclusion
Both converge to same \( L \) \( L \) \( L \) Limit = \( L \)
Converge to different values \( L_1 \) \( L_2 \) DNE
One or both grow unbounded \( \pm \infty \) \( \pm \infty \) DNE (describe as \( \pm \infty \))
Values oscillate wildly No pattern No pattern DNE

Question 1 of 5

Based on the table, what is \( \lim_{x \to 3} g(x) \)?

Question 2 of 5

What is \( \lim_{x \to 1} h(x) \)?

Question 3 of 5

A table shows that as \( x \to 4^{-} \), \( f(x) \to 10 \), and as \( x \to 4^{+} \), \( f(x) \to 10 \). Also, \( f(4) = 7 \). What is \( \lim_{x \to 4} f(x) \)?

Question 4 of 5

What is \( \lim_{x \to 2} p(x) \)?

Question 5 of 5

If a table of values shows \( f(x) \) approaching \( 3.000 \) from both sides as \( x \to 5 \), but \( f(5) \) is undefined, what can you conclude?