EVALUATING LIMITS ALGEBRAICALLY

Evaluate the following limits without using a calculator.

Problem 1 :

lim x 3 2x2 - 5x - 3x - 3

Solution :

Given, lim x 3 2x2 - 5x - 3x - 3= lim x 3 (x - 3) (2x + 1)x - 3= lim x 3 2x + 1= 2(3) + 1= 7

Problem 2 :

lim x 2 x4 - 16x - 2

Solution :

Given, lim x 2 x4 - 16x - 2= lim x 2 x22 - 42x - 2= lim x 2 x2 + 4 x2 - 4x -2= lim x 2 x2 + 4 x2 - 22x -2= lim x 2 x2 + 4 (x + 2) (x - 2)x -2= lim x 2 x2 + 4 (x + 2)= 22 + 4 (2 + 2)= (4 + 4) × 4= 32

Problem 3 :

lim x -1 x4 + 3x3 - x2 + x + 4x + 1

Solution : 

Given, lim x -1 x4 + 3x3 - x2 + x + 4x + 1x4 + 3x3 - x2 + x + 4 ÷ (x + 1)
long-division-2
= lim x -1 (x + 1) x3 + 2x2 -3x + 4x + 1= lim x -1 x3 + 2x2 -3x + 4= (-1)3 + 2(-1)2 -3(-1) + 4= -1 + 2 + 3 + 4= 8

Problem 4 :

lim x 0 x + 4 - 2x

Solution :

Given, lim x 0 x + 4 - 2x = lim x 0 x + 4 - 2x × x + 4 + 2x + 4 + 2= lim x 0 (x + 4 - 4)xx + 4 + 2= lim x 0 xxx + 4 + 2= lim x 0 1x + 4 + 2= 10 + 4 + 2= 12 + 2= 14

Problem 5 :

lim x 3 x + 6 - xx - 3

Solution :

Given, lim x 3 x + 6 - xx - 3 = lim x 3 x + 6 - xx - 3 × x + 6 + xx + 6 + x= lim x 3 x + 6 - x2 (x - 3) x + 6 + x= lim x 3 -(x - 3) (x + 2)(x - 3)x + 6 + x= lim x 3 -(x + 2)x + 6 + x= -(3 + 2)3 + 6 + 3= -53 + 3= -56

Problem 6 :

lim x -2 12 + 1xx + 2

Solution :

Given, lim x -2 12 + 1xx + 2 = lim x -2 x + 22xx + 2 = lim x -2 x + 22x · 1x + 2= lim x -2 12x= 12(-2)= -14

Problem 7 :

lim x 12 x-1 - 2x - 12

Solution :

Given, lim x 12 x-1 - 2x - 12 = lim x 12 1x - 2x - 12 = lim x 12 1 - 2xx 2x - 12 = lim x 12 2(1 - 2x)x(2x - 1) = lim x 12 -2(2x - 1)x(2x - 1) = lim x 12 -2x= -212= -2 × 2= -4

Problem 8 :

lim x 3 1x2 - 19x - 3

Solution :

Given,lim x 3 1x2 - 19x - 3 = lim x 3 9 - x29x2 x - 3 = lim x 3 -x2 - 99x2 x - 3 = lim x 3 -x2 - 329x2 x - 3 = lim x 3 -(x + 3) (x - 3)9x2 x - 3 = lim x 3 -x + 39x2= -3 + 39(3)2= -681= -227

Problem 9 :

lim x 0 5x2 + x - 5x

Solution :

Given, lim x 0 5x2 + x - 5x= lim x 0 5x - 5x2 + xxx2 + x = lim x 0 5x - 5x2 - 5xx3 + x2 = lim x 0 -5x2x2(x + 1) = lim x 0 -5x + 1 = -50 + 1 = -5

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