Mathematics

Problem

Let

F(x)=x\Bigl[(1+\tfrac{2}{x})^{x}-e^{2}\Bigr], \qquad x>0.

The factor

x

amplifies a small difference that emerges as

x

grows large. a. Express

F(x)

in a form that involves an exponential function whose exponent is the natural logarithm of

(1+\tfrac{2}{x})^{x}

. Show how this reformulation isolates a product of

x

with a difference that approaches

0

. b. Evaluate

\displaystyle \lim_{x\to\infty} F(x)

. Show every major step, including any series expansions, limit laws, or algebraic simplifications you use. c. Suppose

H(x)=x[(1+\tfrac{a}{x})^{x}-e^{a}]

for a positive constant

a

. Generalize your work in parts (a) and (b) to find

\displaystyle \lim_{x\to\infty} H(x)

in terms of

a

, and briefly justify why your result holds for all

a>0