Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Determining Horizontal Asymptotes of Rational Functions, Exercises of Calculus

How to find the horizontal asymptotes of a rational function by evaluating the limit of the function as x approaches infinity. It includes examples and general rules for polynomial functions.

Typology: Exercises

2021/2022

Uploaded on 09/12/2022

heathl
heathl ๐Ÿ‡บ๐Ÿ‡ธ

4.5

(11)

237 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
.
HORIZONTAL ASYMPTOTES
In order to determine the horizontal asymptotes of a rational function R(x) = f(x)
g(x)we must evaluate
the following limit: lim
xโ†’โˆž
R(x).
Example: Evaluate lim
xโ†’โˆž
2x2+xโˆ’1
x2+ 1
Solution: Notice that 2x2+xโˆ’1โ†’ โˆž and x2+ 1 โ†’ โˆž as xโ†’ โˆž. For this function we can multiply
the numerator and denominator by 1
x2to determine the limit.
lim
xโ†’โˆž
2x2+xโˆ’1
x2+ 1 = lim
xโ†’โˆž
(2x2+xโˆ’1) (1
x2)
(x2+ 1) (1
x2)= lim
xโ†’โˆž
2 + 1
xโˆ’1
x2
1 + 1
x2
=
lim
xโ†’โˆž
2 + lim
xโ†’โˆž
1
xโˆ’lim
xโ†’โˆž
1
x2
lim
xโ†’โˆž
1 + lim
xโ†’โˆž
1
x2
=2+0โˆ’0
1+0 = 2
Here we are exploiting the fact that lim
xโ†’โˆž
1
xn= 0, where n > 0 is a rational number. Therefore
f(x) = 2x2+xโˆ’1
x2+ 1 has a horizontal asymptote along the line y= 2 as xapproaches โˆž.
Example: Find the horizontal asymptotes of g(x) = x+ 1
x2+ 3xโˆ’4
lim
xโ†’โˆž
x+ 1
x2+ 3xโˆ’4= lim
xโ†’โˆž
(x+ 1) (1
x2)
(x2+ 3x+ 4) (1
x2)= lim
xโ†’โˆž
1
x+1
x2
1 + 3
x2โˆ’4
x2
=
lim
xโ†’โˆž
1
x+ lim
xโ†’โˆž
1
x
lim
xโ†’โˆž
1 + lim
xโ†’โˆž
3
x2โˆ’lim
xโ†’โˆž
4
x2
=0+0
1 + 0 โˆ’0= 0
We conclude that g(x) has a horizontal asymptote along the line y= 0.
Note: In general for polynomial functions f(x) and g(x) we multiply the numerator and denominator
by 1
xmwhere mis the highest power of xthat appears in the denominator.
pf2

Partial preview of the text

Download Determining Horizontal Asymptotes of Rational Functions and more Exercises Calculus in PDF only on Docsity!

HORIZONTAL ASYMPTOTES

In order to determine the horizontal asymptotes of a rational functionthe following limit: lim R(x) = f g^ ((xx)) we must evaluate xโ†’โˆž R(x). Example: Evaluate lim xโ†’โˆž^2 x^2 x^ + (^2) + 1^ x^ โˆ’^1 Solution: Notice that 2the numerator and denominator byx^2 + x โˆ’ 1 โ†’ โˆž 1 and x^2 + 1 โ†’ โˆž as x โ†’ โˆž. For this function we can multiply x^2 to determine the limit. xlimโ†’โˆž^2 x^2 x^ +^2 + 1^ x^ โˆ’ 1 = lim xโ†’โˆž^ (2x^2 +^ x^ โˆ’^ 1)

( (^) x 12 ) (x^2 + 1)^ (^ x^12 )^ = lim^ xโ†’โˆž

2 +^1 x โˆ’ (^) x^12 1 + x^12 =^ xlimโ†’โˆž^ 2 + lim x limโ†’โˆž^ x 1 + limโ†’โˆž x^1 xโ†’โˆž^ โˆ’^ x xlim 1 โ†’โˆž 2 x^12 = 2 + 01 + 0^ โˆ’ 0 = 2 Here we are exploiting the fact that (^) xlimโ†’โˆž x^1 n = 0, where n > 0 is a rational number. Therefore f (x) =^2 x^2 x^ + (^2) + 1^ x^ โˆ’ 1 has a horizontal asymptote along the line y = 2 as x approaches โˆž. Example: Find the horizontal asymptotes of g(x) = (^) x (^2) + 3x^ + 1x โˆ’ 4

xlimโ†’โˆž x^2 + 3^ x^ + 1x โˆ’ 4 = lim xโ†’โˆž^ (x^ + 1)

( (^) x 12 ) (x^2 + 3x + 4)^ (^ x^12 )^ = lim^ xโ†’โˆž

(^1) x + x (^12) 1 + x^32 โˆ’ (^) x^42 = x limโ†’โˆž 1 + lim^ xlimโ†’โˆž xโ†’โˆž^1 x^ + lim x 3 x 2 โ†’โˆž โˆ’ (^) x^1 xlimโ†’โˆž x 42 = (^) 1 + 00 + 0 โˆ’ 0 = 0 We conclude that g(x) has a horizontal asymptote along the line y = 0. Noteby 1 : In general for polynomial functions f (x) and g(x) we multiply the numerator and denominator xm^ where^ m^ is the highest power of^ x^ that appears in the denominator.

Given the rational function R(x) = f g^ ((xx) )= (^) bamnxxmn^ ++^ abnmโˆ’โˆ’^11 xxnmโˆ’โˆ’^1 1 + (^) +^ ยท ยท ยท ยท ยท ยท^ + +^ a b^11 xx^ + +^ a b^00 , the horizontal asymptotes for R(x) are determined by the following cases: (i) If deg(f (x)) > deg(g(x)), R(x) has no horizontal asymptote. (ii) If deg(f (x)) = deg(g(x)), R(x) has a horizontal asymptote at y = (^) bamn. (iii) If deg(f (x)) < deg(g(x)), R(x) has a horizontal asymptote at y = 0. Note: deg(f (x)) and deg(g(x)) represent the degree of f (x) and g(x), respectively. Examples (i) R(x) = x x^32 + 1โˆ’ 1 , deg(f (x)) > deg(g(x)). R(x) has no horizontal asymptote.

(ii) y = R 4 (x) =^4 x^22 โˆ’x (^2 3) + 7x^ + 1 , deg(f (x)) = deg(g(x)). R(x) has an horizontal asymptote along the line 2 = 2. (iii) R(x) = (^) xx (^2) + 5+ 1x, deg(f (x)) < deg(g(x)). R(x) has an horizontal asymptote along the line y = 0.