 # Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. lim x->oo (x−3)/(sqrt 4x^2+25) illusiia 2021-09-25 Answered

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. $$\displaystyle\lim_{{x}→∞}\frac{{{x}−{3}}}{{√{4}{x}^{{2}}+{25}}}$$

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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. $$\displaystyle\lim_{{x}→∞}\frac{{{x}−{3}}}{{√{4}{x}^{{2}}+{25}}}:\frac{{x}}{{x}}=\lim_{{x}→∞}\frac{{{1}-{\left(\frac{{3}}{{x}}\right)}}}{{√{\left({4}+{\left(\frac{{25}}{{x}^{{2}}}\right)}\right)}}}=\frac{{{1}-{0}}}{{√{4}+{0}}}=\frac{{1}}{{2}}$$