Mathematical Methods · calculus of exponential functions · manipulative

Which base makes the limit exactly 1?

For every base a the quotient (ah − 1)/h settles on some level as h → 0. Slide a until that level lands on 1 — only one base can do it.

(ah − 1)/h target 1 limit level

y = (ah − 1)/h near h = 0

Limit level L(a) = ln a

Slide the base a — the open circle is the value the quotient is heading for

Base a

2

Step size h

0.01

Estimates as h shrinks

h(ah − 1)/h

Working