Ratio of polynomials at infinity

Statement

Calculate $\displaystyle\lim_{x\to+\infty}\frac{3x^2-x+1}{5x^2+2}$.

Solution

An $\infty/\infty$ form. Numerator and denominator have the same degree ($2$): we factor out $x^2$: $\frac{3x^2-x+1}{5x^2+2} = \frac{x^2\left(3-\frac{1}{x}+\frac{1}{x^2}\right)}{x^2\left(5+\frac{2}{x^2}\right)} = \frac{3-\frac{1}{x}+\frac{1}{x^2}}{5+\frac{2}{x^2}}.$ As $x\to+\infty$ the terms with $1/x$ and $1/x^2$ tend to $0$, so the limit is the ratio of the leading coefficients: $\lim_{x\to+\infty}\frac{3x^2-x+1}{5x^2+2} = \boxed{\frac{3}{5}}.$

Links

Topics: Limits
Concepts: Indeterminate forms · Hierarchy of infinities
Skills: Calculating limits
Exercise type: Limit calculation