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Find the limit as approaches negative infinity.
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Q. Find the limit as approaches negative infinity.
- Identify highest power: Identify the highest power of in both the numerator and the denominator to simplify the limit expression.In the numerator, the highest power of is inside the square root, which is equivalent to when taken outside the square root. In the denominator, the highest power of is .
- Divide by : Divide both the numerator and the denominator by , the highest power of in the denominator.
- Simplify expression: Simplify the expression inside the limit by canceling out the powers of .\[\lim_{x \to -\infty}\left(\frac{\sqrt{\left(\frac{x^{\(8\)}}{x^{\(4\)}}\right)-\left(\frac{\(5\)x^{\(3\)}}{x^{\(4\)}}\right)}}{\left(\frac{\(3\)x^{\(4\)}}{x^{\(4\)}}\right)+\left(\frac{\(4\)}{x^{\(4\)}}\right)}\right) = \lim_{x \to -\infty}\left(\frac{\sqrt{x^{\(4\)}-\frac{\(5\)}{x}}}{\(3\)+\frac{\(4\)}{x^{\(4\)}}}\right)
- Approach negative infinity: As \(x\) approaches negative infinity, the terms \(-\frac{5}{x}\) and \(\frac{4}{x^{4}}\) approach \(0\).\(\lim_{x \rightarrow -\infty}\left(\frac{\sqrt{x^{4}-\frac{5}{x}}}{3+\frac{4}{x^{4}}}\right) = \lim_{x \rightarrow -\infty}\left(\frac{\sqrt{x^{4}}}{3}\right)\)
- Evaluate square root: Evaluate the square root of \(x^4\) as \(x\) approaches negative infinity.\(\newline\)Since \(x\) is approaching negative infinity, we must consider that the square root of \(x^4\) is the absolute value of \(x^2\), which is \(|x^2|\). However, since \(x\) is negative, \(|x^2| = x^2\) because the square of a negative number is positive.\(\newline\)\(\lim_{x \to -\infty}(\sqrt{x^{4}})/3 = \lim_{x \to -\infty}(x^2)/3\)
- Determine limit: Determine the limit of \(x^2/3\) as \(x\) approaches negative infinity.\(\newline\)As \(x\) approaches negative infinity, \(x^2\) approaches positive infinity. Therefore, the limit of \(x^2/3\) as \(x\) approaches negative infinity is positive infinity.\(\newline\)\(\lim_{x \to -\infty}(x^2)/3 = +\infty\)
