On the Limit of the Ratio of Sin X to X

Extract

The fallacies in the traditional approach to the problem of evaluating

in a pre-calculus course have been well brought out in an interesting article by Mr. Turton where the limit is obtained by means of polynomial approximations to the area of a circle. On re-reading the article recently it occurred to me that, at very little extra cost in effort and depth of ideas, a rigorous evaluation of the limit was possible which was completely free from appeals to the physico-spatial properties of geometrical figures and assumptions of symmetry and continuity which Mr. Turton makes.