Find the limit as n approaches infinity of (2n - n^3)/(12 + n^3). The options offered are: no correct answer, 0, 5, and 1.

Step-by-step solution

1. 1

   Identify the dominant power on each side. Both numerator and denominator are cubic in $n$: the numerator's leading term is $-n^{3}$ and the denominator's is $+n^{3}$. Equal degrees mean the limit will be the ratio of those leading coefficients.

2. 2

   Divide every term by n^3. This makes the claim precise rather than asserted:

   $\frac{2n-n^{3}}{12+n^{3}}=\frac{\dfrac{2}{n^{2}}-1}{\dfrac{12}{n^{3}}+1}.$

3. 3

   Send the reciprocal powers to zero. Both $\dfrac{2}{n^{2}}\to 0$ and $\dfrac{12}{n^{3}}\to 0$, leaving

   $\lim_{n\to\infty}\frac{0-1}{0+1}=-1.$

4. 4

   Watch the sign — it is the whole point. The numerator's cubic term is negative, so the sequence approaches $-1$ from below and the answer is negative. Reading $2n-n^{3}$ as if the $n^3$ were positive is what produces the tempting answer $1$.

5. 5

   Match against the offered options. The computed value $-1$ is not $0$, not $5$ and not $1$. The correct choice is therefore A: no correct answer.

6. 6

   Spot-check numerically. At $n=100$ the quotient is $\dfrac{200-1000000}{12+1000000}=-0.9998$, and at $n=1000$ it is $-0.999998$ — clearly converging to $-1$.