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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). lim_(x rarr oo)(sqrt(-24x^(2)+49x^(6)))/(9+7x^(3)+2x^(2)) Answer:

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).\newlinelim⁡x→∞−24x2+49x69+7x3+2x2 \lim _{x \rightarrow \infty} \frac{\sqrt{-24 x^{2}+49 x^{6}}}{9+7 x^{3}+2 x^{2}} \newlineAnswer:

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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).\newlinelim⁡x→∞−24x2+49x69+7x3+2x2 \lim _{x \rightarrow \infty} \frac{\sqrt{-24 x^{2}+49 x^{6}}}{9+7 x^{3}+2 x^{2}} \newlineAnswer:
  1. Factor out highest power: We are asked to find the limit of the function as xx approaches infinity. To simplify the expression, we can factor out the highest power of xx in the numerator and denominator.
  2. Factor out x3x^3: In the numerator, the highest power of xx is x6x^6, so we factor out x3x^3 from the square root to get x3x^3 times the square root of the remaining expression.\newline−24x2+49x6\sqrt{-24x^{2}+49x^{6}} = x3−24x4+49x^3 \sqrt{-\frac{24}{x^{4}} + 49}
  3. Rewrite with factored terms: In the denominator, the highest power of xx is x3x^3, so we factor out x3x^3 from each term.(9+7x3+2x2)=x3×(9x3+7+2x)(9+7x^{3}+2x^{2}) = x^3 \times \left(\frac{9}{x^3} + 7 + \frac{2}{x}\right)
  4. Cancel out x3x^3 terms: Now we rewrite the original limit expression with the factored terms.\lim_{x \to \infty}\left(\frac{\sqrt{\(-24\)x^{\(2\)}+\(49\)x^{\(6\)}}}{\(9\)+\(7\)x^{\(3\)}+\(2\)x^{\(2\)}}\right) = \lim_{x \to \infty}\left(\frac{x^\(3\) \sqrt{\(-24\)/x^{\(4\)} + \(49\)}}{x^\(3\) \left(\(9\)/x^\(3\) + \(7\) + \(2\)/x\right)}\right)
  5. Simplify as \(x\) approaches infinity: We can now cancel out the \(x^3\) terms in the numerator and denominator.\(\newline\)\(\lim_{x \to \infty}\left(\frac{x^\(3\) \sqrt{-\frac{\(24\)}{x^{\(4\)}} + \(49\)}}{x^\(3\) \left(\frac{\(9\)}{x^\(3\)} + \(7\) + \frac{\(2\)}{x}\right)}\right) = \lim_{x \to \infty}\left(\frac{\sqrt{-\frac{\(24\)}{x^{\(4\)}} + \(49\)}}{\frac{\(9\)}{x^\(3\)} + \(7\) + \frac{\(2\)}{x}}\right)
  6. Simplify square root and denominator: As \(x\) approaches infinity, the terms with \(x\) in the denominator approach zero. Therefore, we can simplify the expression by removing those terms.\(\newline\)\[\lim_{x \to \infty}\left(\frac{\sqrt{-\frac{24}{x^{4}} + 49}}{\frac{9}{x^3} + 7 + \frac{2}{x}}\right) = \lim_{x \to \infty}\left(\frac{\sqrt{0 + 49}}{0 + 7 + 0}\right)
  7. Final answer: Now we can simplify the square root and the denominator.\newlinelim⁡x→∞(497)=77\lim_{x \to \infty}\left(\frac{\sqrt{49}}{7}\right) = \frac{7}{7}
  8. Final answer: Now we can simplify the square root and the denominator.\newlinelim⁡x→∞(49)/7=7/7\lim_{x \to \infty}(\sqrt{49})/7 = 7/7Finally, we simplify the fraction to get the final answer.\newline7/7=17/7 = 1

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