Prove equations for asymptotes of standard hyperbola

[JsStewartFan]

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.

 

[QuarkCharmer]

Do you have to use the definition in terms of the limit difference?

 

[JsStewartFan]

By the limit difference, I just meant the definition for the asymptotes that says if the limit as x approaches infinity of (f(x) - (mx + b)) equals zero, then that y=mx+b is an asymptote of the function f(x), which is the hyperbola in question. It's listed under related equations.

I wasn't sure what you meant by liking my whiteboard, but I used MS Word 2007 with Equation Tools to write out the limit expression. Then I copied and pasted. I've never used Latex Reference before, so I used what I knew.

Also, could you help me with navigation? I can't figure out a quick way to get to the homework and course work help section.

Thanks!
Terry

 

[JsStewartFan]

Oh, I read your question wrong. I think since Mr. Stewart only explained slant asymptotes in terms of the limit definition, that he's expecting me to use it here. This is a problem right after that section. I know I can use setting the hyperbola to zero instead of 1 (saw that in another post here), but I should be able to use limits to prove it, too.

Thanks,
Terry

 

[JsStewartFan]

I finally found the homework and course work section under Science, so I'm OK there.
Thanks,
Terry

 

[JsStewartFan]

QuarkCharmer:
I think I've proved this two ways - one I found in Sullivan's Algebra & Trigonometry book (proved in the text), ed. 7, page 793. The other one I proved using the difference of limits as I wanted to, but I had to separate the proof into the four quadrants, since the hyperbola is not a function unless I do that. It turned out OK, I think.

I'd like to send it to you, but I took about 30 minutes after I copied and pasted my new Word document, trying to fix all the spaciing and so forth - then I was logged out automatically and lost it all. Let me know if there's some way I can get the Word document to you.

I'd appreciate you looking at it for me, and maybe you can post it to this question if correct.

Thanks,
Terry