Evaluating a limit of a rational function using limit laws

Since $p(x)$ is a ratio of two polynomials, apply the quotient rule for limits: $\lim_{x\to -1}\frac{2x^3+1}{x^2-5} = \frac{\lim_{x\to -1}(2x^3+1)}{\lim_{x\to -1}(x^2-5)}$, valid because the denominator's limit will turn out to be nonzero. For the numerator, use the sum and power rules: $\lim_{x\to -1}(2x^3+1) = 2(-1)^3+1 = -2+1 = -1$.

For the denominator, again use the sum and power rules: $\lim_{x\to -1}(x^2-5) = (-1)^2-5 = 1-5 = -4$, which is nonzero, so the quotient rule applies. Combining these results gives $\lim_{x\to -1} p(x) = \dfrac{-1}{-4} = \dfrac{1}{4}$.

The quotient rule for limits requires that the limit of the denominator be nonzero before you can divide the two limits. In part a, this was checked by confirming that the limit of the denominator, x squared minus five, evaluated to negative four, which is nonzero. This justified applying the quotient rule to find the overall limit. So the answer is the quotient rule.