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PROBLEM:
Evaluate \displaystyle \lim\limits_{x\to \infty }\left(\frac{4x+5}{x^2+1}\right)
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Solution:
Divide by the highest denominator power
\begin{align*} \lim\limits_{x\to \infty }\left(\frac{4x+5}{x^2+1}\right) & =\lim\limits_{x\to \infty }\left(\frac{4x+5}{x^2+1}\cdot \frac{\displaystyle\frac{1}{x^2}}{\displaystyle\frac{1}{x^2}}\right) \\ \\ &=\lim\limits_{x\to \infty }\left(\frac{\displaystyle\frac{4x}{x^2}+\displaystyle\frac{5}{x^2}}{\displaystyle\frac{x^2}{x^2}+\displaystyle\frac{1}{x^2}}\right)\\ \\ &=\lim\limits_{x\to \infty }\left(\frac{\displaystyle\frac{4}{x}+\displaystyle\frac{5}{x^2}}{1+\displaystyle\frac{1}{x^2}}\right) \\ \\ &=\displaystyle\frac{0+0}{1+0} \\ \\ &=0 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
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