Exercise 8.7

Note that by L’Hospital’s Rule,

If $f(x)=\sin <!--\; FUNCTION\; APPLICATION\; -->x∕x$, then

Letting $g(x)=x\cos <!--\; FUNCTION\; APPLICATION\; -->x-\sin <!--\; FUNCTION\; APPLICATION\; -->x$, the numerator in the expression above for $f^{\prime }(x)$, we have $g^{\prime }(x)=-x\sin <!--\; FUNCTION\; APPLICATION\; -->x\le 0$ for $0\le x\le \pi ∕2$, so that $g(x)$ decreases in this interval from $g(0)=0$ to $g(\pi ∕2)=-1$. Hence $f^{\prime }(x)<0$ for $0<x<\pi ∕2$, so that it decreases steadily in this interval, and its values lie between $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}f(x)=1$ and $f(\pi ∕2)=2∕\pi$.