Calculus begins and ends with limits. Therefore, the College Board loves to test them. However, many students stumble when a function behaves differently on each side of a point. Those special values are called one-sided limits, and they appear in almost every topic that follows: continuity, derivatives, and even integrals.

This article explains what one-sided limits are, how to read them from algebra and graphs, and how to avoid common traps. By the end, students will be able to justify answers confidently on both the AP® Calculus AB and BC exams.

Why Study One-Sided Limits? 

Because a derivative requires the limit to exist from both sides, ignoring one-sided limits is risky. Moreover, many real-world models change rules at a boundary, such as tax brackets or speed bumps.

Left-hand limit vs. right-hand limit:

• Left-hand limit looks at values as $x$ approaches $a$ from the left.
• Right-hand limit looks as $x$ approaches $a$ from the right.

Quick benefits for the AP® exam:

• Confirms continuity quickly.
• Spots jump discontinuities in multiple-choice items.
• Justifies answers in free-response proofs.

Formal Definition and Notation

Mathematicians shorten sentences with symbols. The limit from the left is written

$\displaystyle \lim_{x\to a^-}f(x)$

and the limit from the right is

$\displaystyle \lim_{x\to a^+}f(x).$

Plain-language version:

“f(x) gets arbitrarily close to L when x approaches a only from values smaller (or larger) than a.”

Example #1 (Algebraic/Piecewise)

Problem: Find $\displaystyle \lim_{x\to 3^-}g(x)$ and $\displaystyle \lim_{x\to 3^+}g(x)$ for

$g(x)= \begin{cases}x^2-1,&x<3 \\ 2x+1,&x \ge 3 \end{cases}$

Step-by-step solution:

1. For values a little less than 3, use $x^2-1$.
2. Substitute 3: $3^2-1=8.$ Therefore, $\displaystyle \lim_{x\to 3^-}g(x)=8.$
3. For values a little greater than 3, use $2x+1$.
4. Substitute 3: $2(3)+1=7.$ Therefore, $\displaystyle \lim_{x\to 3^+}g(x)=7.$

Key takeaway: Because 8 ≠ 7, the overall limit at 3 does not exist. Limits from one side can disagree.

Estimating One-Sided Limits from Graphs

Graphs talk if the viewer knows the language. However, subtle clues matter.

Visual cues:

• Open circle → value not included.
• Closed circle → point on the curve.
• Arrow → unbounded direction.
• Dashed line → asymptote.

Practical tips for estimating limit value from graphs:

1. Trace along the curve toward the target x-value.
2. Keep eyes on y-values, not where the pen ends.
3. If the curve shoots upward forever, note “∞” (limit does not exist).

Example #2 (Basic Sketch)

A graph shows a jump at $x=2$. As $x$ approaches 2 from the left, the curve sits near y = 3. From the right, points rest near y = −1.

Find

a) $\displaystyle \lim_{x\to 2^-}f(x)$

b) $\displaystyle \lim_{x\to 2^+}f(x)$

Solution:

a) Moving along the left branch, y-values head to 3. Therefore, $\lim_{x\to 2^-}f(x)=3.$

b) Moving along the right branch, y-values head to −1. Therefore, $\lim_{x\to 2^+}f(x)=-1.$

Because the numbers differ, the two one-sided limits reveal a jump discontinuity. Consequently, the overall limit does not exist.

Scale Matters: Hidden Behavior on Graphs

Graphing calculators and apps choose default windows. However, zooming out can mask danger zones. Issues of scale simply mean that what looks smooth at one view may be wild up close.

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

Example #3 (Rational Function)

• Window A: [−10,10] for both axes shows a straight line through (1,2).
• Window B: [0.5,1.5] for x and [0,5] for y reveals a hole at $x=1$.

Explanation: The algebra simplifies to $h(x)=x+1$ when $x\ne1$, but $x=1$ is excluded. The large window blurs the missing point. Therefore, always zoom in before trusting a limit read-off.

Hidden traps on scale: a hole may appear filled, a vertical asymptote may look like a gap, and oscillations might flatten.

When a Limit Does Not Exist

Three classic reasons:

1. Unbounded behavior – the function goes to ±∞.

2. Infinite oscillation – values swing faster and never settle.
3. Mismatch – left- and right-hand limits give different numbers.

Mini-Example #4 (Unbounded)

$\displaystyle \lim_{x\to 0^+}\frac{1}{x}=\infty$

Since values grow without bound, the limit DNE.

Mini-Example #5 (Oscillation)

$\displaystyle \lim_{x\to 0}\sin\bigl(\tfrac{1}{x}\bigr)$ does not exist because the sine term oscillates between −1 and 1 infinitely often near 0.

Therefore, spotting DNE quickly saves precious exam time.

Quick Reference Chart: Vocabulary & Symbols

Term	Symbol	Short Definition	Visual Tip
One-sided limit	$\lim_{x\to a^\pm}$	Limit from left or right only	Arrow pointing to point from one side
Left-hand limit	$\lim_{x\to a^-}$	x approaches a from smaller values	Approach from left
Right-hand limit	$\lim_{x\to a^+}$	x approaches a from larger values	Approach from right
Jump discontinuity	—	Function “jumps” to new y	Two distinct dots stacked
Vertical asymptote	$x=a$	y grows to ±∞ near x = a	Dashed vertical line
Oscillatory behavior	—	Rapid back-and-forth changes	Wave that tightens
DNE	—	Limit does not exist	“×” or gap

Putting It All Together

Checklist for any one-sided limit problem:

1. Identify the side: look for “+” or “−” superscript.
2. Choose the correct branch or graph piece.
3. Substitute cautiously or trace toward the point.
4. Note special cases: hole, asymptote, oscillation.
5. Compare two one-sided limits to decide continuity.

Free-response justification tip: State the numeric result and the reason. For example, “$\lim_{x\to2^-}f(x)=3$ because the y-values of the left branch approach 3.”

Conclusion

Mastering one-sided limits unlocks many later skills. Students now know formal notation, methods for estimating limit value from graphs, and warning signs when limits fail to exist. Moreover, understanding issues of scale prevents costly calculator mistakes.

Keep practicing with mixed problems, and confidence on both the AB and BC exams will rise. After all, every derivative proof starts with a humble limit from one side. Good luck!

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!

• 1.2 Defining Limits and Using Limit Notation
• 1.4 Estimating Limit Values from Tables

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