Differential and Integral Calculus by Feliciano and Uy, Exercise 1.4, Problem 3

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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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