Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{-3 x+x^{4}}}{7+8 x^{2}+10 x}$ $\newline$ Answer:

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Q. Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).

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\lim _{x \rightarrow \infty} \frac{\sqrt{-3 x+x^{4}}}{7+8 x^{2}+10 x}

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Answer:

Analyze Behavior of Functions: To find the limit of the function as $x$ approaches infinity, we need to analyze the behavior of the numerator and the denominator separately. We will look for the highest power of $x$ in both the numerator and the denominator to simplify the expression.
Divide by Highest Power of x: In the numerator, the highest power of x is $x^4$ , and in the denominator, the highest power of x is $x^2$ . To simplify the limit, we can divide both the numerator and the denominator by $x^2$ , the highest power of x in the denominator.
Simplify Expression Inside Square Root: When we divide each term in the numerator and the denominator by $x^2$ , we get: $\lim_{x \rightarrow \infty}(\sqrt{x^4 - 3x}/x^2) / (\frac{7}{x^2} + 8 + \frac{10}{x})$
Ignore Terms Approaching Zero: Now we simplify the expression inside the square root in the numerator by dividing each term by $x^2$ : $\lim_{x \rightarrow \infty}\left(\sqrt{x^2 - \frac{3}{x}}\right) / \left(\frac{7}{x^2} + 8 + \frac{10}{x}\right)$
Simplified Limit Calculation: As $x$ approaches infinity, the terms $-\frac{3}{x}$ and $\frac{7}{x^2}$ in the numerator and the denominator, respectively, will approach zero. The term $\frac{10}{x}$ in the denominator will also approach zero. So we can ignore these terms for the limit calculation.
Square Root of $x^2$ : The simplified form of the limit is now: $\lim_{x \rightarrow \infty}\left(\frac{\sqrt{x^2}}{8}\right)$
Limit Calculation Result: The square root of $x^2$ is $x$ , so the limit simplifies further to: $\newline$ $\lim_{x \rightarrow \infty}\frac{x}{8}$
Limit as $x$ Approaches Infinity: As $x$ approaches infinity, the limit of $\frac{x}{8}$ is also infinity, since the value of $x$ becomes unbounded and there is no other term in the denominator to counteract this growth.
Conclusion: Therefore, the limit of the function as $x$ approaches infinity is infinite, which means the limit does not exist (DNE).