Dividing by $n^2$ to Evaluate Limits of Rational Partial Sums

Conditions

• The expression is a quotient of polynomials in $n$.
• The limit is taken as $n \to \infty$.
• The highest power of $n$ in the denominator is $n^2$.

Reasoning, step by step

1. Expand the denominator to identify the highest power of $n$.
2. Divide both the numerator and the denominator by this highest power ($n^2$).
3. Observe the behavior of each term as $n \to \infty$.
4. Conclude that terms with $n$ in the denominator approach 0, while the simplified numerator grows without bound.

Example

Dividing $\frac{2n^3}{n^2+3n+2}$ by $n^2$ yields $\frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}}$. As $n \to \infty$, $\frac{3}{n} \to 0$ and $\frac{2}{n^2} \to 0$, so the denominator approaches 1 while the numerator $2n \to \infty$.

Common misconceptions

• Believing that dividing by $n^2$ changes the value of the limit.
• Assuming the denominator's growth forces the whole fraction to 0 without checking the numerator's degree.

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