For small values of

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

For small values of xx, many trigonometric expressions can be rewritten so that known limit properties become evident. Consider the function g(x)=sin⁡(5x)−sin⁡(2x)x.g(x)=\frac{\sin(5x)-\sin(2x)}{x}. Answer the following questions. (a) Rewrite g(x)g(x) so that it is expressed entirely as a sum or difference of terms each having the form sin⁡(cx)x\dfrac{\sin(c x)}{x}, where cc is a constant. Show all algebraic steps. (b) Using your result from part (a), determine lim⁡x→0g(x)\displaystyle\lim_{x\to 0}g(x). Justify your work by referencing appropriate limit properties. (c) Explain, in complete sentences, why your result in part (b) is consistent with the behavior of g(x)g(x) near x=0x=0. Include a brief discussion of continuity where appropriate.

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