Limits are the launchpad for both derivatives and integrals. Therefore, a solid grip on limits will power every later topic in AP® Calculus. Today’s spotlight is the graphical and numerical approach to evaluating limits. These twin techniques build intuition and speed, two skills prized on the AP® exam.

Throughout this lesson, phrases such as graphical limits, finding limits graphically, and finding limits algebraically appear often. Keep an eye out for them.

Limits in Plain Language

• A limit tells “where a function is heading” as $x$ moves toward some number $a$.
• A two-sided limit looks at both the left and right approach, while a one-sided limit isolates just one direction.
• Notation refresher: $\displaystyle \lim_{x\to a}f(x).$ If the left‐hand limit and right‐hand limit match, then the limit exists; otherwise, it does not.

Graphical Limits: Reading the Story Told by the Curve

The Visual Toolbox

When analyzing a graph, scan for:

• Open dots (holes) and closed dots (actual points)
• Vertical or horizontal asymptotes
• Jumps where the curve hops to a new height
• Smooth sections that signal continuity

Step-by-Step Example #1 (Piecewise Graph)

Consider the graph of $g(x)$ shown below. (Imagine a filled circle at $(2,3)$ and an open circle at $(2,1)$; the left branch heads to the open circle, and the right branch starts at the filled circle.)

Task: Evaluate $\displaystyle \lim_{x\to 2}g(x).$

Solution:

1. First, find the left-hand limit. As $x$ approaches 2 from the left, $g(x)$ moves toward 1.
2. Next, find the right-hand limit. Approaching from the right, $g(x)$ moves toward 3.
3. Because 1 ≠ 3, the two-sided limit does not exist (DNE).

Quick Tips for Finding Limits Graphically

• Trace the curve with your finger from both sides.
• Check open vs. closed dots carefully.
• Note any vertical asymptote; if the height shoots up or down without bound, expect $\pm\infty$.
• Verify both one-sided limits before declaring DNE.

Numerical Limits: Tables That Talk

Building a “Zoom-In” Table

A numerical or tabular method zooms in on $x=a$ by choosing values like $a-0.1,;a-0.01,;a-0.001$ and their right-hand twins. Watching the $y$ -values settle reveals the limit.

Step-by-Step Example #2 (Rational Function with a Hole)

Investigate $h(x)=\dfrac{x^{2}-1}{x-1}$ near $x=1$.

1. Create a table.

x	0.9	0.99	0.999	1.001	1.01	1.1
h(x)	1.9	1.99	1.999	2.001	2.01	2.1

2. Observe the $y$ -values. They trend toward 2.

3. Therefore, $\displaystyle \lim_{x\to 1}h(x)=2$ even though $h(1)$ is undefined (hole).

Technology Note

A graphing calculator’s tbl feature quickly generates such tables. However, double-check that you are closing in symmetrically from both sides.

Bridging Representations

A function’s graph, table, and algebraic rule all describe the same limit. Being fluent in switching viewpoints is an AP® skill.

Mini Example #3

• Numerical finding: A table shows $f(x)$ approaches 4 as $x\to 3$.
• Verbal statement: “The limit of $f(x)$ as $x$ approaches 3 is 4.”
• Graphical confirmation: On the graph, both branches head toward the point $(3,4)$, though a hole may exist there.

Quick Contrast: Finding Limits Algebraically vs. Graphically/Numerically

Sometimes algebra saves time; other times a visual view prevents errors.

Step-by-Step Example #4

Given $p(x)=\dfrac{x^{2}-9}{x-3}$, find $\displaystyle \lim_{x\to 3}p(x).$

Algebraic Method

1. Factor numerator: $x^{2}-9=(x-3)(x+3)$.
2. Cancel $(x-3)$ (only valid for $x\neq 3$): results in $p(x)=x+3$.
3. Substitute $x=3$: limit equals 6.

Graphical/Numerical Check

Plot or table confirms values near 3 stay near 6. Therefore, algebra and graph agree.

When to choose each:

• Algebra: removable holes, simple rational forms.
• Graphical/Numerical: jump discontinuities, absolute-value pieces, or tricky radicals where algebra feels messy.

Common Pitfalls and How to Dodge Them

• Misreading open vs. closed points: zoom in on the dot style.
• Picking x-values that are not close enough: halve the step size until the y-values stabilize.
• Forgetting one-sided analysis: always test both directions before concluding DNE. Meanwhile, label answers clearly as left-hand, right-hand, or two-sided.

Quick Reference Chart: Must-Know Vocabulary

Term	Definition/Key Feature
Limit	Number $L$ that $f(x)$ approaches as $x$ approaches $a$
One-sided limit	Limit from only left ($x\to a^-$) or right ($x\to a^+$)
Removable discontinuity	Hole that can be “fixed” by redefining a single point
Jump discontinuity	Left-hand and right-hand limits differ; graph literally jumps
Infinite limit	Function grows without bound; notation $\pm\infty$
Continuity	Function is connected with no breaks at $a$
Vertical asymptote	x-value where function shoots to $\pm\infty$
Numerical limit	Estimate based on a table of values
Graphical limits	Limits determined by reading a graph

Conclusion

Graphical and numerical approaches to evaluating limits give rapid insight and a safety net when algebra stalls. Consequently, toggling among tables, graphs, and formulas deepens understanding and raises AP® exam confidence. Next, practice switching representations until each feels natural. Limits will then become an ally, not an obstacle, as calculus adventures continue.

Sharpen Your Skills for AP® Calculus AB-BC

Are you preparing for the AP® Calculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

• 1.8 Determining Limits Using the Squeeze Theorem
• 1.10 Exploring Types of Discontinuities

Need help preparing for your AP® Calculus AB-BC exam?

Albert has hundreds of AP® Calculus AB-BC practice questions, free responses, and an AP® Calculus AB-BC practice test to try out.