a)
f(x)=2x−7x3−4
2x−7=0
2x=7 :2
x=3,5 - a więc funkcja jest ciągła w x=2
x→2lim2x−7x3−4=2⋅2−723−4=4−78−4=−34=−34
b)
x→3limx2−3x9−x2=x→3lim−x(3−x)(3−x)(3+x)= x→3lim−x3+x=−33+3=−2
c)
x→2lim12−3x28−x3=x→2lim3⋅(4−x2)23−x3=x→2lim3⋅(2−x)(2+x)(2−x)(22+2x+x2)=x→2lim3⋅(2+x)(4+2x+x2)=3⋅(2+2)4+2⋅2+22=124+4+4=1212=1
d)
x→−2limx2−x−6x2−2x−8
x2−2x−8=0
Δ1=(−2)2−4⋅1⋅(−8)=4+32=36
x1=22+6=28=4
x2=22−6=−24=−2
x2−x−6=0
Δ2=(−1)2−4⋅1⋅(−6)=1+24=25
x1=21+5=26=3
x2=21−5=−24=−2
x→−2lim(x−3)(x+2)(x−4)(x+2)=x→−2lim(x−3x−4=−2−3−2−4=−5−6=56
e)
x→1lim4x2−7x+3x3−1
x3−1=x3−13=(x−1)(x2+x+1)
4x2−7x+3=0
Δ=(−7)2−4⋅4⋅3=49−48=1
x1=4⋅27−1=86=43
x2=4⋅27+1=88=1
x→1lim4x2−7x+3x3−1=x→1lim4(x−43)(x−1)(x−1)(x2+x+1)=x→1lim4(x−43)x2+x+1=
=4⋅(1−43)1+1+1=4−33=3
f)
x→31lim3x2+2x−19x2−1
9x2−1=(3x−1)(3x+1)=3⋅(x−31)⋅3⋅(x+31)=9⋅(x−31)(x+31)
3x2+2x−1=0
Δ=22−4⋅3⋅(−1)=4+12=16
x1=2⋅3−2−4=6−6=−1
x2=2⋅3−2+4=62=31
x→31lim3x2+2x−19x2−1=x→31lim3(x+1)(x−31)9⋅(x−31)(x+31)=x→31lim(x+1)3⋅(x+31)=
=31+13⋅(31+31)=341+1=2⋅43=23
f)
x→5limx2−4x−525x−x3
25x−x3=x(25 −x2)=x(5−x)(5+x)
x2−4x−5=0
Δ=(−4)2−4⋅1⋅(−5)=16+20=36
x1=2⋅14+6=210=5
x2=2⋅14−6=2−2=−1
x→5limx2−4x−525x−x3=x→5lim(x−5)(x+1)x(5−x)(5+x)=x→5lim(x−5)(x+1)−x(x−5)(5+x)= x→5limx+1−x(5+x)=
=−5+15⋅(5+5)=−65⋅10=−650=−325
h)
x→−4limx2+7x+12x3+3x2−4x
x3+3x2−4x=0
x(x2+3x−4)=0
Δ1=32−4⋅1⋅(−4)=9+16=25
x1=2⋅1−3+5=22=1
x2=2⋅1−3−5=−28=−4
x2+7x+12=0
Δ2=72−4⋅1⋅12=49−48=1
x1=2−7+1=−26=−3
x2=2−7−1=−28=−4
x→−4limx2+7x+12x3+3x2−4x=x→−4lim(x+3)(x+4)x(x−1)(x+4)=x→−4limx+3x(x−1)=
=−4+3−4⋅(−4−1)=−1−4⋅(−5)=−20
i)
x→21lim8x2−6x+116x3−2
16x3−2=2(8x3−1)=2((2x)3−13)=2(2x−1)((2x)2+2x+12)=2(2x−1)(4x2+2x+1)=2⋅2⋅(x−21)(4x2+2x+1)
8x2−6x+1=0
Δ=(−6)2−4⋅8⋅1=36−32=4
x1=2⋅86+2=168=21
x2=2⋅86−2=164=41
x→21lim8x2−6x+116x3−2=x→21lim8(x−21)(x−41)4(x−21)(4x2+2x+1)=x→21lim2(x−41)4x2+2x+1=
=2⋅(21−41)4⋅221+2⋅21+1=1−211+1+1=213=6