For each positive real parameter

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

p

, define the function

L(p)=\lim_{x\to\infty}\left(1+\frac{p}{x}\right)^{x}.

Assume throughout that all limits exist. (a) Show that

L(p)

exists for every

p>0

and determine

L(p)

explicitly in terms of

p

. Provide full justification for the existence of the limit before evaluating it. (b) Treating

L(p)

as a function of the parameter

p

, compute

\frac{d}{dp}L(p)

. Show all steps, including any differentiation rules you employ. (c) Find the value of

p

such that

L(p)=10

. Explain your reasoning and verify that the solution you obtain satisfies the required condition.