Limit of a Sequence

The limit of a rational sequence (the ratio of two polynomials in $n$) as $n\to\infty$ is found by comparing the degrees of the numerator and denominator:

• numerator degree lower → limit $0$,
• degrees equal → ratio of the leading coefficients,
• numerator degree higher → improper limit ($+\infty$ or $-\infty$).

Technique: divide the numerator and denominator by the highest power of $n$; the lower-order terms tend to zero. For example $\lim\dfrac{2n^2+3n-1}{n^2+5}=2$. Enter both polynomials (write powers with ^, e.g. 2n^2 + 3n - 1).