Limits and Derivatives

Limits and Derivatives

Question

Evaluate space space space space space space space space space space space space limit as straight x rightwards arrow square root of 2 of fraction numerator straight x to the power of 9 minus 3 straight x to the power of 8 plus straight x to the power of 6 minus 4 straight x squared minus 16 straight x plus 84 over denominator straight x to the power of 5 minus 3 straight x to the power of 4 minus 4 straight x plus 12 end fraction

Answer

Since at space straight x space equals space square root of 2 comma end root numerator and denominator becomes zero.
∴ it  is of 0 over 0  form.
∴ By factor theorem, space space left parenthesis straight x minus square root of 2 right parenthesis is a factor of both numerator and denominator.
rightwards double arrow space space straight x space plus square root of 2 is also a factor of both numerator and denominator because irrational roots occur in pairs.
So, x-2 is a factor of both numerator and denominator.
By long division method, dividing numerator and denominator by x-2.
x- 3x+ x- 9x- 4x- 16x + 84 = (x-2) (x- 3x+ 2x-5x+ 4x-19x+8x - 42)  and  x5-3x-4x + 12 = (x-2) (x- 3x+ 2x -6)

∴  space space space limit as straight x rightwards arrow square root of 2 of fraction numerator straight x to the power of 9 minus 3 straight x to the power of 8 plus straight x to the power of 6 minus 9 straight x to the power of 4 minus 4 straight x squared minus 16 straight x plus 84 over denominator straight x to the power of 5 minus 3 straight x to the power of 4 minus 4 straight x plus 12 end fraction

space space space space space space equals space space space stack space lim with straight x rightwards arrow square root of 2 below fraction numerator left parenthesis straight x squared minus 2 right parenthesis left parenthesis straight x to the power of 7 minus 3 straight x to the power of 6 plus 2 straight x to the power of 5 minus 5 straight x to the power of 4 plus 4 straight x cubed minus 19 straight x squared plus 8 straight x minus 42 right parenthesis over denominator left parenthesis straight x squared minus 2 right parenthesis left parenthesis straight x cubed minus 3 straight x squared plus 2 straight x minus 6 right parenthesis end fraction

   space space space space space space space space space equals space space fraction numerator left parenthesis square root of 2 right parenthesis end root to the power of 7 minus 3 left parenthesis square root of 2 right parenthesis to the power of 6 plus 2 left parenthesis square root of 2 right parenthesis end root to the power of 5 minus 5 left parenthesis square root of 2 right parenthesis to the power of 4 plus 4 left parenthesis square root of 2 right parenthesis end root cubed minus 19 left parenthesis square root of 2 right parenthesis end root squared plus 8 left parenthesis square root of 2 right parenthesis end root minus 42 over denominator left parenthesis square root of 2 right parenthesis cubed minus 3 left parenthesis square root of 2 right parenthesis squared plus 2 left parenthesis square root of 2 right parenthesis minus 6 end fraction
space space space space space space space space space equals space fraction numerator 8 square root of 2 minus 24 plus 8 square root of 2 minus 20 plus 8 square root of 2 minus 38 plus 8 square root of 2 minus 42 over denominator 2 square root of 2 minus 6 plus 2 square root of 2 minus 6 end fraction space space space space       space space space space open square brackets because space straight x space not equal to square root of 2 space rightwards double arrow space straight x squared minus 2 not equal to 0 close square brackets

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