If a curve is arbitrarily close to an infinite segment of a line L, then L is called an asymptote of the curve. Equivalently, we give the following math homework help solution :

Definition: The line y = mx + c (m ≠ 0) is called an asymptote of a curve y = ƒ(x) if the perpendicular distance of any point P(x, y) on the curve from the line approaches zero as x ∞ + or - ∞.

We shall now determine the conditions in order that the line

y = mx + c

is an asymptote of the curve y = ƒ(x). If p denotes the perpendicular distance of any point P(x, y) on the curve from the line, then

By definition p0 as x ± ∞

limx ± ∞ (y – mx – c) = 0 (i)

Since otherwise the limit in (ii) would be

This determines the value of m. Now, by (i), we have

c = m limx ± ∞ (y – mx) (iv)

This determines the value of c.

Rule: The line y = mx + c (m ≠ 0) is an asymptote of the curve y = ƒ(x), where m and c are determined by

m limx ± ∞ y/x ,c = limx ± ∞ (y – mx).

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