Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{23 x+64 x^{9}}}{7 x^{4}+6 x^{3}}$ $\newline$ 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{23 x+64 x^{9}}}{7 x^{4}+6 x^{3}}

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Answer:

Identify Highest Power: To find the limit of the given function as $x$ approaches infinity, we need to analyze the behavior of the numerator and the denominator separately. We look for the highest power of $x$ in both the numerator and the denominator to simplify the expression.
Simplify by Dividing: In the numerator, the highest power of $x$ inside the square root is $x^9$ , and in the denominator, the highest power of $x$ is $x^4$ . To simplify, we can divide both the numerator and the denominator by $x^4$ , the highest power in the denominator.
Evaluate Terms: When we divide each term by $x^4$ , we get the following: $\newline$ Numerator: $\sqrt{\frac{23x}{x^4} + \frac{64x^9}{x^4}} = \sqrt{\frac{23}{x^3} + 64x^5}$ $\newline$ Denominator: $\frac{7x^4}{x^4} + \frac{6x^3}{x^4} = 7 + \frac{6}{x}$
Remove Terms Approaching Zero: As $x$ approaches infinity, any term with $x$ in the denominator will approach zero. Therefore, $\frac{23}{x^3}$ approaches $0$ and $\frac{6}{x}$ approaches $0$ .
Final Simplification: After removing the terms that approach zero, we are left with: $\newline$ Numerator: $\sqrt{64x^5}$ $\newline$ Denominator: $7$
Simplify Square Root: We can simplify the square root of $64x^5$ as $8x^{5/2}$ , because $\sqrt{64}$ is $8$ and $\sqrt{x^9}$ is $x^{9/2}$ .
Final Limit Calculation: Now we have the simplified form of the limit: $\lim_{x \rightarrow \infty}\left(\frac{8x^{5/2}}{7}\right)$
Limit Does Not Exist: Since $x^{5/2}$ approaches infinity as $x$ approaches infinity, the whole expression approaches infinity. Therefore, the limit does not exist (DNE).