Mathematics

Problem

Let

h(x)=\cos x\,e^{x}

, defined for all real

x

. The expression

R(h)=\frac{h\bigl(\tfrac{\pi}{4}+h\bigr)-h\bigl(\tfrac{\pi}{4}\bigr)}{h},\qquad h\neq0

depends on the small increment

h

. (a) Compute

\displaystyle \lim_{h\to0} R(h)

without simplifying

h(x)

into a single elementary function. (b) Verify that your answer in part (a) equals the result obtained by differentiating

h(x)

with an appropriate rule and then substituting

x=\tfrac{\pi}{4}

. Provide all intermediate steps. (c) Explain why the agreement of the two methods in parts (a) and (b) is mathematically guaranteed for any differentiable function.

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