Ratios of polynomials at infinity

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Compute the following limits at infinity:

• (a) $\displaystyle\lim_{x\to+\infty}\frac{2x^{3}-x+1}{5x^{3}+3}$
• (b) $\displaystyle\lim_{x\to+\infty}\frac{x^{2}+1}{x^{3}-2}$
• (c) $\displaystyle\lim_{x\to+\infty}\frac{4x^{3}-2}{x^{2}+x}$

Solution

Compare the degrees of numerator and denominator. (a) Equal degree: limit $=\dfrac{2}{5}=0.4$.

(b) Numerator degree smaller: limit $=0$.

(c) Numerator degree larger: limit $=+\infty$.

$\boxed{a=\tfrac25=0.4;\ b=0;\ c=+\infty}$