Answer to Question #260154 in Calculus for Alunsina

Question #260154

Find the equations of the vertical and horizontal asymptotes of the graphs of the given rational functions.

a. f(x) = 4x²-3x+2/x²-3x+2

b. f(x) = x²-x-20/x²-7x+10

c. f(x) = 4x²-9/2x²-5x+3

d. f(x) = [(4/x-6)+(2/3)]/x

e. f(x) = (4/x²-1)+(2/x+1)


1
Expert's answer
2021-11-03T17:06:09-0400

a.

f(x)=4x23x+2x23x+2f(x)=\dfrac{4x^2-3x+2}{x^2-3x+2}

x23x+20=>(x1)(x2)0x^2-3x+2\not=0=>(x-1)(x-2)\not=0

Vertical asymptotes: x=1,x=2.x=1, x=2.


limxf(x)=limx4x23x+2x23x+2\lim\limits_{x\to-\infin}f(x)=\lim\limits_{x\to-\infin}\dfrac{4x^2-3x+2}{x^2-3x+2}

=limx4x2/x23x/x2+2/x2x2/x23x/x2+2/x2=\lim\limits_{x\to-\infin}\dfrac{4x^2/x^2-3x/x^2+2/x^2}{x^2/x^2-3x/x^2+2/x^2}

=limx43/x+2/x213/x+2/x2=4+0+01+0+0=4=\lim\limits_{x\to-\infin}\dfrac{4-3/x+2/x^2}{1-3/x+2/x^2}=\dfrac{4+0+0}{1+0+0}=4


limxf(x)=limx4x23x+2x23x+2\lim\limits_{x\to\infin}f(x)=\lim\limits_{x\to\infin}\dfrac{4x^2-3x+2}{x^2-3x+2}

=limx4x2/x23x/x2+2/x2x2/x23x/x2+2/x2=\lim\limits_{x\to\infin}\dfrac{4x^2/x^2-3x/x^2+2/x^2}{x^2/x^2-3x/x^2+2/x^2}

=limx43/x+2/x213/x+2/x2=40+010+0=4=\lim\limits_{x\to\infin}\dfrac{4-3/x+2/x^2}{1-3/x+2/x^2}=\dfrac{4-0+0}{1-0+0}=4


Horizontal asymptote: y=4y=4


b.

f(x)=x2x20x27x+10=(x+4)(x5)(x2)(x5)f(x)=\dfrac{x^2-x-20}{x^2-7x+10}=\dfrac{(x+4)(x-5)}{(x-2)(x-5)}

Vertical asymptote: x=2.x=2.

limxf(x)=limxx2x20x27x+10\lim\limits_{x\to-\infin}f(x)=\lim\limits_{x\to-\infin}\dfrac{x^2-x-20}{x^2-7x+10}

=limxx2/x2x/x220/x2x2/x27x/x2+10/x2=\lim\limits_{x\to-\infin}\dfrac{x^2/x^2-x/x^2-20/x^2}{x^2/x^2-7x/x^2+10/x^2}

=limx11/x20/x217/x+10/x2=1+001+0+0=1=\lim\limits_{x\to-\infin}\dfrac{1-1/x-20/x^2}{1-7/x+10/x^2}=\dfrac{1+0-0}{1+0+0}=1



limxf(x)=limxx2x20x27x+10\lim\limits_{x\to\infin}f(x)=\lim\limits_{x\to\infin}\dfrac{x^2-x-20}{x^2-7x+10}

=limxx2/x2x/x220/x2x2/x27x/x2+10/x2=\lim\limits_{x\to\infin}\dfrac{x^2/x^2-x/x^2-20/x^2}{x^2/x^2-7x/x^2+10/x^2}

=limx11/x20/x217/x+10/x2=10010+0=1=\lim\limits_{x\to\infin}\dfrac{1-1/x-20/x^2}{1-7/x+10/x^2}=\dfrac{1-0-0}{1-0+0}=1



Horizontal asymptote: y=1y=1


c.

f(x)=4x292x²5x+3=(2x+3)(2x3)(x1)(2x3)f(x)=\dfrac{4x^2-9}{2x²-5x+3}=\dfrac{(2x+3)(2x-3)}{(x-1)(2x-3)}

Vertical asymptote: x=1.x=1.

limxf(x)=limx4x292x²5x+3\lim\limits_{x\to-\infin}f(x)=\lim\limits_{x\to-\infin}\dfrac{4x^2-9}{2x²-5x+3}

=limx4x2/x29/x22x²/x25x/x2+3/x2=\lim\limits_{x\to-\infin}\dfrac{4x^2/x^2-9/x^2}{2x²/x^2-5x/x^2+3/x^2}

=limx49/x225/x+3/x2=402+0+0=2=\lim\limits_{x\to-\infin}\dfrac{4-9/x^2}{2-5/x+3/x^2}=\dfrac{4-0}{2+0+0}=2




limxf(x)=limx4x292x²5x+3\lim\limits_{x\to\infin}f(x)=\lim\limits_{x\to\infin}\dfrac{4x^2-9}{2x²-5x+3}

=limx4x2/x29/x22x²/x25x/x2+3/x2=\lim\limits_{x\to\infin}\dfrac{4x^2/x^2-9/x^2}{2x²/x^2-5x/x^2+3/x^2}

=limx49/x225/x+3/x2=4020+0=2=\lim\limits_{x\to\infin}\dfrac{4-9/x^2}{2-5/x+3/x^2}=\dfrac{4-0}{2-0+0}=2



d.

f(x)=4x6+23x=12+2x123x(x6)=2x3x(x6)f(x)=\dfrac{\dfrac{4}{x-6}+\dfrac{2}{3}}{x}=\dfrac{12+2x-12}{3x(x-6)}=\dfrac{2x}{3x(x-6)}

Vertical asymptote: x=6.x=6.

limxf(x)=limx4x6+23x\lim\limits_{x\to-\infin}f(x)=\lim\limits_{x\to-\infin}\dfrac{\dfrac{4}{x-6}+\dfrac{2}{3}}{x}

=limx23(x6)=0=\lim\limits_{x\to-\infin}\dfrac{2}{3(x-6)}=0



limxf(x)=limx4x6+23x\lim\limits_{x\to\infin}f(x)=\lim\limits_{x\to\infin}\dfrac{\dfrac{4}{x-6}+\dfrac{2}{3}}{x}

=limx23(x6)=0=\lim\limits_{x\to\infin}\dfrac{2}{3(x-6)}=0


Horizontal asymptote: y=0y=0


e.


f(x)=4x21+2x+1=4+2x2(x+1)(x1)f(x)=\dfrac{4}{x^2-1}+\dfrac{2}{x+1}=\dfrac{4+2x-2}{(x+1)(x-1)}

=2(x+1)(x+1)(x1)=\dfrac{2(x+1)}{(x+1)(x-1)}

Vertical asymptote: x=1.x=1.

limxf(x)=limx(4x21+2x+1)\lim\limits_{x\to-\infin}f(x)=\lim\limits_{x\to-\infin}(\dfrac{4}{x^2-1}+\dfrac{2}{x+1})

=limx2(x+1)(x+1)(x1)=0=\lim\limits_{x\to-\infin}\dfrac{2(x+1)}{(x+1)(x-1)}=0


limxf(x)=limx(4x21+2x+1)\lim\limits_{x\to\infin}f(x)=\lim\limits_{x\to\infin}(\dfrac{4}{x^2-1}+\dfrac{2}{x+1})

=limx2(x+1)(x+1)(x1)=0=\lim\limits_{x\to\infin}\dfrac{2(x+1)}{(x+1)(x-1)}=0

Horizontal asymptote: y=0y=0



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