Let be the function defined for all by q(x)= x -1 [4] x -1

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

Problem

Let qq be the function defined for all x>1x > 1 by q(x)=x−1x4−1.q(x)=\dfrac{\sqrt{x}-1}{\sqrt[4]{x}-1}. The ratio involves radicals of different orders. (a) Determine lim⁡x→1+q(x)\displaystyle\lim_{x\to 1^{+}}q(x), showing all algebraic work. (b) Evaluate lim⁡x→1+q(x)−2x−1\displaystyle\lim_{x\to 1^{+}}\dfrac{q(x)-2}{x-1} and explain why your answer establishes that qq is differentiable at x=1x=1. (c) Using the definition of continuity, state whether qq can be redefined at x=1x=1 to make it continuous. Clearly justify your reasoning. (d) Summarize the key algebraic obstacles that must be overcome when treating limits involving mixed radicals such as those appearing in qq.

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