A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists.Show that each of the following functions has a horizontal asymptote by calculating the given limit.-13xlimx-0o 6 2x-13/29х — 2limx--0o x3 + 10xx2- 8x 13012lim-1/155 — 15х2x->0ox2 + 6хlim702xх->оVx2 6xlim07- 2х

Question
Asked Oct 31, 2019
A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists.
Show that each of the following functions has a horizontal asymptote by calculating the given limit.
-13x
lim
x-0o 6 2x
-13/2
9х — 2
lim
x--0o x3 + 10x
x2- 8x 13
0
12
lim
-1/15
5 — 15х2
x->0o
x2 + 6х
lim
7
0
2x
х->о
Vx2 6x
lim
0
7- 2х
help_outline

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A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists. Show that each of the following functions has a horizontal asymptote by calculating the given limit. -13x lim x-0o 6 2x -13/2 9х — 2 lim x--0o x3 + 10x x2- 8x 13 0 12 lim -1/15 5 — 15х2 x->0o x2 + 6х lim 7 0 2x х->о Vx2 6x lim 0 7- 2х

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Expert Answer

Step 1
-13x
The given function is lim-
x-62
-13x
lim
x-62x
x -13
lim
6
x
- +2
-13
= lim
X 6
+2
x
-13
0+2
13
2
13
-13x
-13x
Since lim
exist, the function
2
has a horizontal
x-o62x
6+ 2x
asymptote
help_outline

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-13x The given function is lim- x-62 -13x lim x-62x x -13 lim 6 x - +2 -13 = lim X 6 +2 x -13 0+2 13 2 13 -13x -13x Since lim exist, the function 2 has a horizontal x-o62x 6+ 2x asymptote

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Step 2
9х—2
The given function is lim
хх +10х —12
9x
2
9x - 2
х
lim
lim
10x-12
10x 12
2
2.
x
= lim
10 12
1+
0-0
1+0-0
0
1
=0
9х —2
9х — 2
=0 exist, the function
3
Since lim
x 10x -12
horizontal asymptote.
has a
x +10x 12
help_outline

Image Transcriptionclose

9х—2 The given function is lim хх +10х —12 9x 2 9x - 2 х lim lim 10x-12 10x 12 2 2. x = lim 10 12 1+ 0-0 1+0-0 0 1 =0 9х —2 9х — 2 =0 exist, the function 3 Since lim x 10x -12 horizontal asymptote. has a x +10x 12

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Step 3
x2-8x13
The given function is lim
5-15x2
8x 13
x2-8x-13
lim
5-15x2
lim
x r
5
-15.
8
1
13
х*
x
=lim
5
-15
1-0-0
0-15
1
15
x2-8x13
x-8x-13
1
exist, the function
15
Since lim
has a horizontal
5-15x
5-15x2
asymptote
help_outline

Image Transcriptionclose

x2-8x13 The given function is lim 5-15x2 8x 13 x2-8x-13 lim 5-15x2 lim x r 5 -15. 8 1 13 х* x =lim 5 -15 1-0-0 0-15 1 15 x2-8x13 x-8x-13 1 exist, the function 15 Since lim has a horizontal 5-15x 5-15x2 asymptote

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Calculus