How do continue/end this infinite limit expression? Shall I use the Reciprocal Rule? - Almost finished

Question

The picture of my work is here

I am almost done with this, but I do not know how to end it? Shall I use the Reciprocal rule?

Answer 1

The final step is to write it as $$ \frac{\lim_{x \to \infty}2+\frac{4}{x}+\frac{1}{x^3}}{\lim_{x \to \infty}5+\frac{2}{x^2}}=\frac{2}{5} \, . $$ If you were doing things completely rigorously, then you would also have to justify why $$ \lim_{x \to \infty}2+\frac{4}{x}+\frac{1}{x^2}=2 $$ and likewise for the denominator. One approach would be to use the sum rule: $$ \lim_{x \to \infty}2+\frac{4}{x}+\frac{1}{x^2}=\lim_{x \to \infty}2+\lim_{x \to \infty}\frac{4}{x}+\lim_{x \to \infty}\frac{1}{x^2} $$ and then use the formal definition of a limit. For instance, $\lim_{x \to\infty}\frac{1}{x^2}=0$ if and only if $$ \forall \varepsilon > 0 \exists N>0 : x>N \implies \left|\frac{1}{x^2}\right|<\varepsilon $$ If we take $N=1/\sqrt{\varepsilon}$, then \begin{align} &x > 1/\sqrt{\varepsilon} \\ \implies & 1/x < \sqrt{\varepsilon} \\ \implies & 1/x^2 < \varepsilon \end{align} as required.