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QuestionFind the limit: lim⁡x→−∞x6−3x42x2−2x+1\lim _{x \rightarrow-\infty} \frac{x^{6}-3 x^{4}}{2 x^{2}-2 x+1}.

Studdy Solution

STEP 1

Assumptions1. We are finding the limit as x approaches negative infinity. . The function is x6−3x4x−x+1\frac{x^{6}-3 x^{4}}{ x^{}- x+1}

STEP 2

We divide the numerator and the denominator by x6x^{6}, the highest power of x in the numerator.
lim⁡x→−∞x6−x42x2−2x+1=lim⁡x→−∞1−x22x4−2x5+1x6\lim{x \rightarrow-\infty} \frac{x^{6}- x^{4}}{2 x^{2}-2 x+1} = \lim{x \rightarrow-\infty} \frac{1-\frac{}{x^{2}}}{\frac{2}{x^{4}}-\frac{2}{x^{5}}+\frac{1}{x^{6}}}

STEP 3

As x approaches negative infinity, the terms with x in the denominator will approach0.
lim⁡x→−∞1−3x22x−2x5+1x6=1−00−0+0\lim{x \rightarrow-\infty} \frac{1-\frac{3}{x^{2}}}{\frac{2}{x^{}}-\frac{2}{x^{5}}+\frac{1}{x^{6}}} = \frac{1-0}{0-0+0}

STEP 4

Calculate the limit.
1−00−0+0=10\frac{1-0}{0-0+0} = \frac{1}{0}The limit as x approaches negative infinity of the function x6−3x42x2−2x+1\frac{x^{6}-3 x^{4}}{2 x^{2}-2 x+1} is undefined because we cannot divide by zero.

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