Evaluate the limit of (5x^3 - 2x + 1)/(3x^3 + 7) as x approaches infinity.

Step-by-step solution

1. 1

   Diagnose the indeterminate form. As $x\to\infty$ the numerator and denominator both grow without bound, giving $\tfrac{\infty}{\infty}$. That is not a value — it only tells you the expression must be rewritten before the limit can be read off.

2. 2

   Divide numerator and denominator by the highest power present. Both are cubic, so divide every term by $x^3$ (legitimate for large $x$, where $x\neq 0$):

   $\frac{5x^{3}-2x+1}{3x^{3}+7}=\frac{5-\dfrac{2}{x^{2}}+\dfrac{1}{x^{3}}}{3+\dfrac{7}{x^{3}}}.$

3. 3

   Send each reciprocal power to zero. For any $k>0$, $\dfrac{1}{x^{k}}\to 0$ as $x\to\infty$, so the three small terms disappear:

   $\lim_{x\to\infty}\frac{5-0+0}{3+0}=\frac{5}{3}.$

4. 4

   State the general rule this illustrates. For a rational function, compare the degrees: equal degrees give the ratio of leading coefficients (here $\tfrac53$), a smaller numerator degree gives $0$, and a larger numerator degree gives $\pm\infty$. The lower-order terms $-2x$, $+1$ and $+7$ never affect the answer.

5. 5

   Check numerically. At $x=10^{6}$ the quotient is

   $\frac{5\cdot 10^{18}-2\cdot 10^{6}+1}{3\cdot 10^{18}+7}=1.6666666\ldots,$

   agreeing with $\tfrac53=1.6\overline{6}$ to many decimal places.