Mathematics

Mathematics

Problem

Let α\alpha denote an angle measured in degrees and define f(α)=sin⁡ααf(\alpha)=\dfrac{\sin \alpha}{\alpha}. (a) Evaluate lim⁡α→0f(α)\displaystyle\lim_{\alpha\to 0} f(\alpha), clearly indicating how you convert between degree and radian measure in your calculation. (b) Explain why the limit in part (a) differs from lim⁡x→0sin⁡xx\displaystyle\lim_{x\to 0} \dfrac{\sin x}{x} when xx is measured in radians. Discuss the implications of this difference for the use of trigonometric limits in calculus and for the small-angle approximations common in physics and engineering.

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