Question

lim√2. x 9 - 3x8 + x6 - 9 x4 - 4 x2 - 16x +84
x5 - 3x4 - 4x + 12

Answer 1

limx→√2 [(x9 – 3x8 + x6 – 9x4 – 4x2 – 16x + 84)/(x5 – 3x4 – 4x + 12)]

Here x – √2 is factor of numerator and denominator both as coefficients are rational, x + √2 must be other factor.

∴ (x + √2)(x – √2) = x2 – 2 is also a factor.

∴ limx→√2 [{(x2 – 2)(x7 + 3x6 + 2x5 – 5x4 + 4x3 – 19x2 + 8x – 42)}/{(x2 – 2)(x3 – 3x2 + 2x – 6)}]

= [(33√2 – 124)/(4√2 – 12)]

= [{(32√2 – 124)(4√2 + 12)}/(32 – 144)]

= [(256 + 384√2 – 496√2 – 1488)/(– 112)]

= [(– 1232 – 112√2)/(– 112)]

= 11 + √2

Answer 2

Answer:

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