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A function is said to vanish to order at if the limit lim→()(−) exists (and is not infinite). The order of vanishing of a function quantifies the rate at which ()→0 as →; for example, if vanishes to order 3 at 2 then () approaches zero at least as quickly as does (−2)3 as →2.

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Answer:

The answer is "vanishes to order 3 at a=1"

Step-by-step explanation:

Please find the complete question in the attached file.

[tex]\to f(x) =\frac{1}{x} -x^2-3x+3\\\\\to f(x)= \frac{1-x^3+3x^2-3x}{x}\\\\\to f(x)= \frac{(1-x)^3}{x} = - \frac{(x-1)^3}{x}\\\\\to \lim_{x \to 1} f(x) = \lim_{x \to 1} \frac{(1-x)^3}{x}[/tex]

vanishes to order 3 at a=1.

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