Define the function for and extend continuously at .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define the function

f(x)=\dfrac{e^{x}-1}{x}

for

x\neq0

and extend

f

continuously at

x=0

. (a) Obtain the Maclaurin series for

f(x)

, writing all terms through

x^{4}

inclusive. (b) Using your series from part (a), determine

\displaystyle \lim_{x\to0} f(x)

and explain how the result is consistent with the derivative of

e^{x}

at the origin. (c) Verify that the series you found in part (a) is the Maclaurin series of

f

by demonstrating term-by-term division or any other valid justification.