- #1

JsStewartFan

- 15

- 0

## Homework Statement

Prove the equation(s) for the asymptotes of a standard hyperbola. That is, prove that the asymptotes for the hyperbola x^2/a^2 - y^2/b^2 = 1 are

y = -(b/a)x and

y = (b/a)x

where foci are at (c,0) and (-c,0); vertices are at (a,0) and (-a,0); difference in distances between the two foci and any point on the hyperbola is a constant 2a or -2a; and

c^2 = a^2 + b^2.

## Homework Equations

If lim(x→∞) [f(x)- (mx+b) ]= 0, then y= mx + b is the equation of an asymptote of f(x).

## The Attempt at a Solution

Using the definition of the asymptote in terms of the limit difference in (2) above, I focused on the first quadrant of the hyperbola and came up with this sequence of equations, but since (-b) does not equal zero, I have an error somewhere:

lim(x→∞)[b/a (x-a)+ bx/a]

= lim(x→∞) [(bx-ba)/a + bx/a]

=lim(x→∞) [(bx-ba+bx)/a]

=lim(x→∞) [(-ba)/a]

= (-b)lim(x→∞)1

= (-b)• 1

= -b ≠ 0

By the way, this is in James Stewart's Calculus - Early Transcendentals, 5th edition, section 4.5, problem 67, page 324. There's no answer in the back of the book. I am reviewing hyperbolas for a high school teacher certification test and got off on this "tangent" because I like to derive or prove equations instead of just memorizing them (when I have time).

Thanks for your help.