Mathematics

Problem

Consider the following limits evaluated at the indicated point. (I)

\displaystyle \lim_{x\to 0}\frac{\tan x}{x}

(II)

\displaystyle \lim_{x\to 0}\frac{1-\cos x}{x}

(III)

\displaystyle \lim_{x\to\infty}\frac{3x^2+7}{x^2-5}

(IV)

\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

0/0

\infty/\infty

. Give your answer by listing the Roman numerals of all qualifying limits.

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