Let S(x)= 1+2x -1-x x^ 2 x

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let S(x)=1+2x−1−xx2sin⁡x,S(x)=\frac{\sqrt{1+2x}-1-x}{x^{2}\sin x}, where xx is measured in radians. (1) Algebraically transform S(x)S(x) into a form that separates the radical expression from the trigonometric factor, so that each may be analyzed by a standard limit theorem. Display every algebraic manipulation. (2) Determine lim⁡x→0S(x)\displaystyle \lim_{x\to 0} S(x). Provide a justification that cites any standard limits or theorems you invoke.

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