Consider the expression

MATHEMATICS · Subjective · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the expression f(x)=sin⁡2(5x)+cos⁡2(3x)−1f(x)=\sin^{2}(5x)+\cos^{2}(3x)-1. (a) Show that −1≤f(x)≤1-1\le f(x)\le 1 for all real xx. (b) Using the result from part (a), evaluate lim⁡x→∞sin⁡2(5x)+cos⁡2(3x)−1x2\displaystyle \lim_{x \to \infty} \dfrac{\sin^{2}(5x)+\cos^{2}(3x)-1}{x^{2}}.

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