Calculus: Analytic Geometry and Calculus, with Vectors

(^138) Functions, limits, derivatives
in which a and b are positive constants, is a hyperbola.
(x, y(x)) lies on the hyperbola and y(x) > 0, then
Y(x) =ba x - a2.
Show that
lim 1b x - y(x)] = 0
z-4. a
Show that if the point
and hence that the line having the equation y = (b/a)x is an asymptote of the
hyperbola. Hint: The formula
x- z2-a2
=x- x2-a2x-} x2-a2

1 x+ xs-as

turns out to be a useful source of information.

8 Find the equations of the asymptotes of thegraphs of the equations

(a) y

'+2

x-1 Ans.: x^1 ,y=1

\2

C

x+1

(b) Y= x 2) A

(r)Y=x+x

ns.: x

Ans.: x

2,y

0,y=x

is

(d) Y = Cx +x Ans.: x = 0

(e) xy = x + y Ans.:x=1,y=1

(f) x2+ y2= x + Y Ans.: None

(g) Y= x-}-1-Vx- Ans. y = 0

9 According to part f of Problem 1, the first of the statements

lim Vx- = 0 (?),

is true.

lim -Vx = 0 (?)

Is the second statement also true?

J 10 Prove that

Remark: This remark is dedicated to unfortunate individuals who never knew

or have forgotten that if

then

S.=1+ 2 + (^3) + 4 +... +(n-1)+n,
S.=n+(n-1)+(n-2)+(n-3)+ + 2 +1
and addition gives 2S. = n(n + 1), so S.= n(n + 1)/2.
(^11) Starting with the definition
(1) n!= 1.2.3...n,