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The asymptotes of $\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1$ Show that the vertical distance between the line $y=(b / a) x$ and the upper half of the right-hand branch $y=(b / a) \sqrt{x^{2}-a^{2}}$ of the hyperbola $\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1$ approaches 0 by showing that$$\lim _{x \rightarrow \infty}\left(\frac{b}{a} x-\frac{b}{a} \sqrt{x^{2}-a^{2}}\right)=\frac{b}{a} \lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-a^{2}}\right)=0.$$Similar results hold for the remaining portions of the hyperbola and the lines $y=\pm(b / a) x.$

Please refer to the solution attached.

Calculus 2 / BC

Chapter 11

Parametric Equations and Polar Coordinates

Section 6

Conic Sections

Parametric Equations

Polar Coordinates

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So in this problem they talk about how the Assam totes distance from the line. Very, and what they want to show is that it approaches zero. So they tell us to show that Miss Limit on the board approaches zero. So let's go ahead and do that. So any time you have something where it's like something minus the square root or screw minus something Um, the way to start is to multiply by the contra git of this so topping bottom. And in doing that, we would end up with X squared minus X squared, minus a squared all over. And then we shall have our conjugal ended denominator. So the X plus X squared minus a square square rooted. Now, notice how the X rays there comes out, and then the negatives air will cancel, so we'll have be over a limit as X approaches. Infinity of base, where over X plus expert, minus a squared, all square rooted. Now a square is a constant, so that doesn't matter it all. So all we need to think about is well, what happens in the denominator? Well, X squared goes to infinity. If we subtract a constant from it still goes towards infinity square rooting that would still go to infinity and then X also goes to infinity. So we have infinity plus infinity. So this should really be over a a squared something going to infinity. And well, in this case, that would just be zero. So you showed what they wanted. So we keep our little proof box and smiling face because we're glad that we're done.

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