Mathematics

Problem

Define

h(x)=\dfrac{\sqrt{1+\sqrt{x}}-1}{x},\quad x>0

. The expression involves a composition of two radical terms. (a) Evaluate

\displaystyle\lim_{x\to 0^{+}}h(x)

and show every algebraic step that justifies your result. (b) Suppose

H(x)=\sqrt{1+\sqrt{x}}

. Use your work in part (a) to interpret the limit you found as an instantaneous rate of change of

H

at a particular point. State the point and interpret the value in the context of the function

H