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Today's calculation of Integral 727

Source: 2000 Asahikawa Medical University entrance exam

June 27, 2011

calculusintegrationgeometrylimitlogarithmscalculus computations

Problem Statement

For positive constant

a

, let

C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})

. Denote by

l(t)

the length of the part

a\leq y\leq t

for

C

and denote by

S(t)

the area of the part bounded by the line

y=t\ (a<t)

and

C

. Find

\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.