COMPUTING LIMITS ALGEBRAICALLY

No calculator is allowed. Evaluate each limit algebraically.

Problem 1 :

x03x+5

Solution:

=x03x+5=3(0)+5=0+5=5

Problem 2 :

x52-3x

Solution:

=x52-3x=2-3(5)=2-15=-13

Problem 3 :

x88-x

Solution:

=x88-x=8-8=0

Problem 4 :

x3x+2x

Solution:

=x3x+2x=3+23=53

Problem 5 :

x0x+2x

Solution:

=x0x+2x=0+20=20=undefined

Problem 6 :

x4x2-16x-4

Solution:

=x4x2-16x-4=x4(x+4)(x-4)x-4=x4(x+4)=4+4=8

Problem 7 :

x2x3-82-x

Solution:

=x2x3-82-x=x2x3-232-xa3-b3=(a-b)a2+b2+ab=x2(x-2)x2+2x+4(2-x)=x2-x2+2x+4=-22+2(2)+4=-(4+4+4)=-12

Problem 8 :

x2x2+2x-8x-2

Solution:

=x2x2+2x-8x-2=x2(x-2)(x+4)(x-2)=x2(x+4)=2+4=6

Problem 9 :

x42x2-5x-12x-4

Solution:

=x42x2-5x-12x-4=x4(2x+3)(x-4)(x-4)=x4(2x+3)=2(4)+3=11

Problem 10 :

x-5x+5x3+125

Solution:

=x-5x+5x3+125=x-5x+5x3+53a3+b3=(a+b)a2+b2-ab=x-5(x+5)(x+5)x2-5x+25=x-51x2-5x+25=1(-5)2-5(-5)+25=175

Problem 11 :

x3x3-3x2-4x+12x-3

Solution:

=x3x3-3x2-4x+12x-3=x3x3-3x2-(4x-12)(x-3)=x3x2(x-3)-4(x-3)(x-3)=x3x2-4(x-3)(x-3)=x3x2-4=32-4=9-4=5

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