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Question

Find the limits :
i. limx1[x2+1x+100]
ii. limx2[x34x2+4xx24]
iii. limx2[x24x34x2+4x]
iv. limx2[x32x2x25x+6]
v. limx1[x2x2x1x33x2+2x]

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Solution

(i) Given : limx1[x2+1x+100]
Substituting x=1
=12+11+100
=1+1101
=2101
limx1[x2+1x+100]=2101

(ii) Given : limx2[x34x2+4xx24]
Substituting the given value,
limx2[x34x2+4xx24]=234×22+4×2224
=816+844
=00
Since it is in 00 form, we have to simplify it.

limx2[x34x2+4xx24]
=limx2x(x24x+4)x2(2)2
=limx2x(x2+2222x)(x2)(x+2)
=limx2x(x2)2(x2)(x+2) (x2)
=limx2x(x2)x+2
Substituing x=2
=2(22)2+2=2(0)4=04=0

(iii) Given : limx2[x24x34x2+4x]
Substituting the given value,
limx2[x24x34x2+4x]=224234×22+4×2
=44816+8
=00
Since it is in 00 form, we have to simplify it.

limx2[x24x34x2+4x]
=limx2[x2(2)2x(x24x+4)]
=limx2[(x2)(x+2)x(x2+2222x)]
=limx2[(x2)(x+2)x(x2)2]
=limx2[x+2x(x2)] (x2)
Substituing x=2
=2+22(22)=42(0)=40
=
Which is not defined.

(iv) Given : limx2[x32x2x25x+6]
Substituting the given value,
limx2[x32x2x25x+6]=232×22225×2+6
=88410+6
=00
Since it is in 00 form, we have to simplify it.

limx2[x32x2x25x+6]
=limx2x2(x2)x23x2x+6
=limx2(x2(x2)x(x3)2(x3))
=limx2(x2(x2)(x2)(x3))
=limx2(x2x3) (x2)
Substituting x=2
=(2)223
=41
=4
limx2[x32x2x25x+6]=4

(v) Given : limx1[x2x2x1x33x2+2x]
=limx1[x2x(x1)1x(x23x+2)]
=limx1[x2x(x1)1x(x(x2)1(x2))]
=limx1[x2x(x1)1x(x1)(x2)]
=limx1[(x2)(x2)1x(x1)(x2)]
=limx1[(x2)212x(x1)(x2)]
=limx1[(x21)(x2+1)x(x1)(x2)]
=limx1[(x3)(x1)x(x1)(x2)] (x1)
=limx1[x3x(x2)]
Substituing x=1
=131(12)
=21×1
=2
limx1[x2x2x1x33x2+2x]=2

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