---
title: "Define and classify asymptotes of a curve"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptote-define-types"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptote-define-types
site: BCT Engineering
---

# Define and classify asymptotes of a curve

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2073 Chaitra
- Video: [ASYMPTOTES | Short Revision Of ASYMPTOTES | Differential Calculus Revision By @gajendrapurohit​](https://www.youtube.com/watch?v=aC3beQFariY) — Dr.Gajendra Purohit - GATE / IIT JAM / CSIR NET
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptote-define-types
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State the types of asymptotes to a curve.

## Short answer

Types include horizontal (y = k), vertical (x = h), and oblique (y = mx + c, m ≠ 0). Also by inspection and for polar curves where r → ∞.

## Answer

An **asymptote** of a curve is a straight line such that the distance of a point on the curve from the line tends to zero as the point goes to infinity along the curve. It is the tangent to the curve at infinity.

#### Types
1. **Asymptote parallel to the x-axis** (horizontal): a line y = k. It is found by equating to zero the coefficient of the highest power of x in the equation of the curve.
2. **Asymptote parallel to the y-axis** (vertical): a line x = h. It is found by equating to zero the coefficient of the highest power of y.
3. **Oblique asymptote** (inclined): a line y = mx + c with m ≠ 0, ∞. The slopes m are the roots of φₙ(m) = 0, where φₙ is the highest-degree part of the equation, and c follows from φₙ₋₁(m) + c φₙ′(m) = 0.

Besides these, a curve may have **asymptotes by inspection** (for example a hyperbola's asymptotes differ from the curve only by a constant) and, in advanced work, circular or curvilinear asymptotes. For polar curves, an asymptote is found from r → ∞ at the angle where 1/f(θ) = 0.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. An asymptote parallel to the x-axis is found by equating to zero:
   A. the coefficient of the highest power of x
   B. the constant term
   C. phi_n(m)
   D. the coefficient of the highest power of y
   Answer: A. Rule for horizontal asymptotes. (Key point: Horizontal)

2. An asymptote parallel to the y-axis is found from:
   A. c = 0
   B. coefficient of the highest power of y
   C. phi_n'(m)
   D. highest power of x
   Answer: B. Rule for vertical asymptotes. (Key point: Vertical)

3. For oblique asymptotes y = mx + c, the slopes m satisfy:
   A. phi_{n-1}(m) = 0
   B. m = c
   C. phi_n(m) = 0
   D. m = 0
   Answer: C. Highest-degree part equated to zero. (Key point: Oblique slope)

4. For a simple root m, c follows from:
   A. phi_{n-1}(m) + c phi_n'(m) = 0
   B. c = m
   C. phi_n'' = 0
   D. phi_n(m) = c
   Answer: A. Standard method. (Key point: Intercept)

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