Which of the following explains why ?

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Which of the following explains why lim⁡x→∞5x2+12x5−7=0\displaystyle \lim_{x \to \infty} \frac{5x^{2}+1}{2x^{5}-7}=0? (a) Both polynomials possess the same highest power of xx, hence the limit equals the ratio of their leading coefficients, 52\dfrac{5}{2}. (b) The numerator tends to a finite number while the denominator grows without bound, therefore the fraction increases without bound and the limit is ∞\infty. (c) The degree of the denominator is higher than that of the numerator, so after dividing by the highest power of xx in the denominator every surviving term contains 1xn\dfrac{1}{x^{n}} with n>0n>0, forcing the limit to 00. (d) Because 2x5−72x^{5}-7 changes sign as xx increases, the fraction alternates between positive and negative values, so no finite limit exists.

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