<s>[INST] Let $R(x) := \frac {3x^4+5x+2} {2x^4+3x^2+7}$. Prove $R$ has a limit for $x \rightarrow \infty$. Let $$R(x) := \frac {3x^4+5x+2} {2x^4+3x^2+7}$$
Prove $R$ has a limit for $x \rightarrow \infty$.
Using a calculator I've expaneded $\frac {3x^4+5x+2} {2x^4+3x^2+7} = \frac {-9x^2}{2(x^4+3x^2+7)} +\frac {5x}{2x^4+3x^2+7} -  \frac {17}{2(x^4+3x^2+7)} + 3/2$.
And I can easily see that three of these fractions has limit $0$ for $x \rightarrow \infty$.
However am I relying a bit to much on a calculator here ? And should I prove for each of these fractions rigorous that the limit is $0$ ? 
 [/INST] You didn't need a calculator for that.
Divide both sides by $x^4$. Now you have expressed it in terms of a function of $1/n$ which goes to zero 0.
This is a calculation method. Now you can use the limit to prove $|R(x)-3/2|<\epsilon$ for $x>N$. This now involves expressing the inequality and proving it's true.
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