Geometrically, β«abβx1βdx means the area under the curve
x1β from a to b, where 0<a<b, and this area gives a positive
number. Using this area argument, in this expository note, we present some
visual representations of some classical results. For examples, we demonstrate
an area argument on a generalization of Euler's limit
(nββlimβ(n(n+1)β)n=e).
Also, in this note, we provide an area argument of the inequality ba<ab,
where eβ€a<b, as well as we provide a visual representation of an
infinite geometric progression. Moreover, we prove that the Euler's constant
Ξ³β[21β,1) and the value of e is near to 2.7. Some parts
of this expository article has been accepted for publication in Resonance -
Journal of Science Education, The Mathematical Gazette, and International
Journal of Mathematical Education in Science and Technology.Comment: 10 pages, 15 figure