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Question

limxx(arctanx+1x+2arctanxx+2) is equal to

A
12
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B
12
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C
1
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D
D.N.E
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Solution

The correct option is B 12
limxx(arctan(x+1x+2)arctan(xx+2))=limxx⎜ ⎜ ⎜ ⎜ ⎜arctan⎜ ⎜ ⎜ ⎜ ⎜x+1x+2xx+21+x(x+1)(x+2)2⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟ ⎟ ⎟=limxx(arctan((x+2)2x2+5x+4))=limxarctan(x+22x2+5x+4)(1x)
=limx1(1+(x+2)2(2x2+5x+4)2)(2x2+5x+4)(x+2)(4x+5)(2x2+5x+4)2(1x2) [By L Hospital's Rule]
=limx(2x2+5x+44x213x10)(2x2+5x+4)2+(x+2)2(x2)=limx2x4+8x3+6x2(2x2+5x+4)2+(x+2)2=limx2+8x+6x2(2+5x+4x2)2+(1x+2x2)2=2+0+0(2+0+0)2+(0+0)=12

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