Short Calculus (MATH 1142)
The Number e
1. We define the number e by the following limit
eh − 1
= 1.
h→0
h
We observe that e is not the evaluation of this limit, but rather the number that makes this
limit evaluate to 1.
lim
(a) Use a calculator to estimate the values of the following two limits to two decimal places
2.7h − 1
h→0
h
lim
and
2.8h − 1
.
h→0
h
lim
(b) What can we conclude about the value of e from these two estimates?
2. For this problem, let f (x) = ex and g(x) = xe .
(a) What types of functions are f and g?
(b) What are the derivatives of f and g, respectively?
(c) Compare the differentiation techniques used to compute f 0 (x) and g 0 (x).
(d) For large values of x, which grows more quickly: f or g? (You may use your calculator
to compute examples.)