a.
f(x)=x2−3x+24x2−3x+2
x2−3x+2=0=>(x−1)(x−2)=0 Vertical asymptotes: x=1,x=2.
x→−∞limf(x)=x→−∞limx2−3x+24x2−3x+2
=x→−∞limx2/x2−3x/x2+2/x24x2/x2−3x/x2+2/x2
=x→−∞lim1−3/x+2/x24−3/x+2/x2=1+0+04+0+0=4
x→∞limf(x)=x→∞limx2−3x+24x2−3x+2
=x→∞limx2/x2−3x/x2+2/x24x2/x2−3x/x2+2/x2
=x→∞lim1−3/x+2/x24−3/x+2/x2=1−0+04−0+0=4
Horizontal asymptote: y=4
b.
f(x)=x2−7x+10x2−x−20=(x−2)(x−5)(x+4)(x−5) Vertical asymptote: x=2.
x→−∞limf(x)=x→−∞limx2−7x+10x2−x−20
=x→−∞limx2/x2−7x/x2+10/x2x2/x2−x/x2−20/x2
=x→−∞lim1−7/x+10/x21−1/x−20/x2=1+0+01+0−0=1
x→∞limf(x)=x→∞limx2−7x+10x2−x−20
=x→∞limx2/x2−7x/x2+10/x2x2/x2−x/x2−20/x2
=x→∞lim1−7/x+10/x21−1/x−20/x2=1−0+01−0−0=1
Horizontal asymptote: y=1
c.
f(x)=2x²−5x+34x2−9=(x−1)(2x−3)(2x+3)(2x−3) Vertical asymptote: x=1.
x→−∞limf(x)=x→−∞lim2x²−5x+34x2−9
=x→−∞lim2x²/x2−5x/x2+3/x24x2/x2−9/x2
=x→−∞lim2−5/x+3/x24−9/x2=2+0+04−0=2
x→∞limf(x)=x→∞lim2x²−5x+34x2−9
=x→∞lim2x²/x2−5x/x2+3/x24x2/x2−9/x2
=x→∞lim2−5/x+3/x24−9/x2=2−0+04−0=2
d.
f(x)=xx−64+32=3x(x−6)12+2x−12=3x(x−6)2x Vertical asymptote: x=6.
x→−∞limf(x)=x→−∞limxx−64+32
=x→−∞lim3(x−6)2=0
x→∞limf(x)=x→∞limxx−64+32
=x→∞lim3(x−6)2=0
Horizontal asymptote: y=0
e.
f(x)=x2−14+x+12=(x+1)(x−1)4+2x−2
=(x+1)(x−1)2(x+1)
Vertical asymptote: x=1.
x→−∞limf(x)=x→−∞lim(x2−14+x+12)
=x→−∞lim(x+1)(x−1)2(x+1)=0
x→∞limf(x)=x→∞lim(x2−14+x+12)
=x→∞lim(x+1)(x−1)2(x+1)=0
Horizontal asymptote: y=0
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