Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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and
Extrema and the Geometric Applications of a Derivative [Ch. 3
Urn
then the straight line y = k z x + b^ is an asymptote (a left inclined asymptote
or, when fe 2 = 0, a left horizontal asymptote). The graph of the function y = f(x)
(we assume the function is single-valued) cannot have more than one right
(inclined or horizontal) and more than one left (inclined or horizontal) asymptote.
Example 1. Find the asymptotes of the curve
Solution. Equating
lotos-
the denominator to zero, we get two vertical asyinpx=
1 and x=l.
We seek the inclined asymptotes. For x > + oo we obtain
b l
k lim = lim
l
*-+o> v }^x z
=- lim (// x) = lim
*
-/
\ \
X 2
Fig. 32
x y^2
S
~l,
hence, the straight line y = x is the right asymptote. Similarly, when* oo,
we have
fc a = Hm ~= 1;
fc = lim
AC->~
Thus, the left asymptote Is y= -x (Fig. 32). Testing a curve for asymptotes
is simplified if we take into consideration the symmetry of the curve.
Example 2. Find the asymptotes of the curve
=0,