Define . The expression involves a composition of two radical…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Define h(x)=1+x−1x,x>0h(x)=\dfrac{\sqrt{1+\sqrt{x}}-1}{x},\quad x>0. The expression involves a composition of two radical terms. (a) Evaluate lim⁡x→0+h(x)\displaystyle\lim_{x\to 0^{+}}h(x) and show every algebraic step that justifies your result. (b) Suppose H(x)=1+xH(x)=\sqrt{1+\sqrt{x}}. Use your work in part (a) to interpret the limit you found as an instantaneous rate of change of HH at a particular point. State the point and interpret the value in the context of the function HH.

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