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The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide nu

Rational functions
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asked 2021-06-26

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. \(\displaystyle\lim_{{x} \rightarrow \infty}\frac{{{x}−{3}}}{{\sqrt{{4}{x}^{{2}}+{25}}}}\)

Answers (1)

2021-06-27

Highest power of x in denominator is \(\displaystyle{x}^{{2}}\), but since it is beneath square root, we are going to divide both numenator and denominator with x. \(\displaystyle\lim_{{x} \rightarrow \infty}\frac{{{x}−{3}}}{{\sqrt{{4}{x}^{{2}}+{25}}}}:\frac{{x}}{{x}}=\lim_{{x} \rightarrow \infty}\frac{{{1}-{\left(\frac{{3}}{{x}}\right)}}}{{\sqrt{{\left({4}+{\left(\frac{{25}}{{x}^{{2}}}\right)}\right)}}}}=\frac{{{1}-{0}}}{{\sqrt{{4}+{0}}}}=\frac{{1}}{{2}}\)

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