Consider the following limits evaluated at the indicated point

MATHEMATICS · Subjective · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Consider the following limits evaluated at the indicated point. (I) lim⁡x→0tan⁡xx\displaystyle \lim_{x\to 0}\frac{\tan x}{x} (II) lim⁡x→01−cos⁡xx\displaystyle \lim_{x\to 0}\frac{1-\cos x}{x} (III) lim⁡x→∞3x2+7x2−5\displaystyle \lim_{x\to\infty}\frac{3x^2+7}{x^2-5} (IV) lim⁡x→1ln⁡xx−1\displaystyle \lim_{x\to 1}\frac{\ln x}{x-1} Indicate every limit among (I)–(IV) whose initial form qualifies for direct evaluation by a theorem that applies specifically to ratios of functions approaching 0/00/0 or ∞/∞\infty/\infty. Give your answer by listing the Roman numerals of all qualifying limits.

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