---
title: "Vertical Tangent Line — AP Calc Definition & Exam Guide"
description: "A vertical tangent line occurs where a curve is continuous but its slope becomes infinite, so the derivative is undefined. Key for differentiability questions in AP Calc."
canonical: "https://fiveable.me/ap-calc/key-terms/vertical-tangent-line"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 2"
---

# Vertical Tangent Line — AP Calc Definition & Exam Guide

## Definition

A vertical tangent line is a tangent with undefined slope that occurs where a function is continuous but the limit of the difference quotient is infinite, making the function non-differentiable at that point even though the graph has no break, corner, or cusp.

## What It Is

A vertical tangent line is exactly what it sounds like: the [tangent line](/ap-calc/key-terms/tangent-line "fv-autolink") at a point on a curve is perfectly vertical. Since slope is rise over run and a vertical line has zero run, the slope is undefined. In calculus language, the [limit](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/defining-convergent-divergent-infinite-series/study-guide/CIVFHStGQM90EJ4GtIDB "fv-autolink") of the difference quotient at that point is +∞ or −∞, so f'(x) does not exist there.

Here's the part that trips people up. The [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") is still **continuous** at a vertical tangent. The graph flows smoothly through the point with no jump or hole. The classic example is f(x) = x^(1/3) at x = 0. The cube root curve passes right through the origin, but it gets steeper and steeper as you approach x = 0, and the tangent line snaps to vertical. So a vertical tangent is one of the famous ways a function can be continuous but NOT differentiable. The other two are corners (like |x|) and cusps. You can also see vertical tangents in implicit curves like circles, where x² + y² = 25 has vertical tangents at (5, 0) and (−5, 0).

## Why It Matters

Vertical tangents live in [Unit 2](/ap-calc/unit-2 "fv-autolink") (Differentiation: Definition and Fundamental Properties), specifically the idea that differentiability implies continuity but continuity does NOT imply differentiability. The vertical tangent is your go-to counterexample. The concept comes back hard in Unit 3 with implicit differentiation, where finding vertical tangents means finding where the denominator of dy/dx equals zero (while the numerator doesn't). For BC, it shows up again in Unit 9 with parametric curves, where a vertical tangent happens when [dx/dt](/ap-calc/key-terms/dxdt "fv-autolink") = 0 and dy/dt ≠ 0. So this one idea threads through three units. If you understand why the slope blows up to infinity, all three versions are the same question wearing different outfits.

## Connections

### Derivative & Differentiability (Unit 2)

The derivative is the [slope of the tangent line](/ap-calc/key-terms/slope-of-the-tangent-line "fv-autolink"), so a vertical tangent means the derivative doesn't exist at that point. A vertical tangent is the 'infinite slope' failure of differentiability, as opposed to the 'two different slopes' failure you get at a corner.

### Implicit Differentiation (Unit 3)

When you implicitly differentiate something like a circle or ellipse, [dy/dx](/ap-calc/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD "fv-autolink") usually comes out as a fraction. Vertical tangents happen where the denominator is zero and the numerator isn't. This is one of the most common ways the exam actually asks you to find a vertical tangent.

### Parametric and Polar Curves (Unit 9, BC only)

For a [parametric curve](/ap-calc/key-terms/parametric-curve "fv-autolink"), dy/dx = (dy/dt)/(dx/dt). A vertical tangent occurs when dx/dt = 0 while dy/dt ≠ 0. Same logic as implicit differentiation: zero on the bottom, nonzero on top.

### Absolute Value Functions (Unit 2)

f(x) = |x| at x = 0 is the other classic non-differentiable point, but for a different reason. A corner has two different finite slopes meeting; a vertical tangent has one slope racing off to infinity. Both are continuous, neither is differentiable.

## On the AP Exam

No released FRQ uses the phrase 'vertical tangent line' as its headline, but the concept is a multiple-choice staple. The most common stem shows a graph or piecewise function and asks 'at which point is f continuous but not differentiable?' A vertical tangent is one of the right-answer options, alongside corners and cusps. The other big appearance is in implicit differentiation problems: after you find dy/dx, you may be asked to find all points where the tangent line is vertical. Your move is to set the denominator of dy/dx equal to zero, make sure the numerator isn't also zero there, and confirm the point is actually on the original curve. BC students should expect the parametric version, where dx/dt = 0 signals a vertical tangent. In every version, the underlying skill is recognizing what 'undefined slope' means analytically.

## Vertical Tangent Line vs Vertical Asymptote

Both involve vertical lines, but they're nearly opposites. At a vertical tangent, the function is CONTINUOUS and the curve actually touches the point (think x^(1/3) at x = 0); only the slope blows up. At a vertical asymptote, the function itself blows up to ±∞ and is discontinuous there (think 1/x at x = 0); the curve never reaches the line. Quick test: can you plug the x-value into the function and get a real output? If yes, it might be a vertical tangent. If the function value is undefined or infinite, you're looking at an asymptote.

## Key Takeaways

- A vertical tangent line has undefined slope, so the derivative does not exist at that point even though the function is continuous there.
- Vertical tangents are one of the three classic ways a function can be continuous but not differentiable, along with corners and cusps.
- The standard example is f(x) = x^(1/3) at x = 0, where the curve passes smoothly through the origin but the tangent line is vertical.
- In implicit differentiation problems, find vertical tangents by setting the denominator of dy/dx equal to zero while keeping the numerator nonzero.
- For BC parametric curves, a vertical tangent occurs where dx/dt = 0 and dy/dt ≠ 0.
- Don't confuse a vertical tangent with a vertical asymptote; at a vertical tangent the function is continuous, while at an asymptote the function is discontinuous.

## FAQs

### What is a vertical tangent line in calculus?

It's a tangent line with undefined slope, occurring where the limit of the difference quotient is +∞ or −∞. The function is continuous at that point but not differentiable, like f(x) = x^(1/3) at x = 0.

### Is a function differentiable at a vertical tangent?

No. Differentiability requires a finite slope, and a vertical tangent has infinite slope. The function is still continuous there, which makes vertical tangents a perfect counterexample showing continuity does not imply differentiability.

### What's the difference between a vertical tangent and a vertical asymptote?

At a vertical tangent the curve actually passes through the point and the function is continuous; only the slope is infinite. At a vertical asymptote the function value itself goes to ±∞ and the function is discontinuous, like 1/x at x = 0.

### How do you find vertical tangent lines with implicit differentiation?

After finding dy/dx as a fraction, set the denominator equal to zero and the numerator not equal to zero, then check that the resulting points satisfy the original equation. For x² + y² = 25, dy/dx = −x/y, so vertical tangents occur where y = 0, at the points (5, 0) and (−5, 0).

### Is a vertical tangent the same as a cusp or corner?

No, though all three make a function non-differentiable while staying continuous. At a corner (like |x| at x = 0), two different finite slopes meet. At a vertical tangent, the slope approaches infinity from both sides in the same direction. A cusp also has infinite slopes, but they approach from opposite directions.

## Related Study Guides

- [Unit 2 Overview: Differentiation](/ap-calc/unit-2/review/study-guide/HGq8OPntbFvqp1uRuEcj)

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