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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that the curve r = 2 − csc θ (also a conchoid) has the line y = −1 as a horizontal asymptote by showing that limr→±∞ y = −1. Use this fact to help sketch the conchoid.

To determine

To find: The polar curve r=2cscθ has the line y=1 as horizontal asymptote by showing that limr±y=1 .

Explanation

Given:

The polar curve equation r=2cscθ

Calculation:

The polar curve is r=2cscθ (1).

The equation for the variable y is given below,

y=rsinθyr=sinθcscθ=ry

Substitute (ry) for (cscθ) in equation (1),

r=2(ry)r2=(ry)yr=12ry=r(2r)y=1(2r1)

Take limit as r±

limr±x=limr±y=1(2r1)=1(2limr±1r1)=1(01)=1

Hence, x=1 is a vertical asymptote

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