Question

(a) How is the number e defined? e is the number such that lim h → 0 e−h − 1 h = 1. e is the number such that lim h → 0 eh − 1 h = 1. e is the number such that lim h → [infinity] eh + 1 h = 1. e is the number such that lim h → 0 eh + 1 h = 1. e is the number such that lim h → 1 e−h − 1 h = 1. (b) Use a calculator to estimate the values of the following limits to two decimal places. lim h → 0 2.7h − 1 h = lim h → 0 2.8h − 1 h = What can you conclude about the value of e? 0 < e < 1 2.7 < e < 2.8 0.99 < e < 2.7 0.99 < e < 1.03 1.03 < e < 2.7

Answer 1

Answer:

is the number such that lim h → [infinity] eh + 1 h = 1.

Step-by-step explanation:

The number e is known as the Euler's constant or Euler's number. The number  basically is a mathematical expression that is equal to the rational number 2.71828. In addition, it is also the base of the natural logarithm or the Naperian logarithms. It is the number that the natural logarithm is equal to one. In addition, it is the limit of :

$\lim_{n \to \infty} (1 + \frac{1}{n })^{n}$

as n approaches infinity.

The expression is calculated as a sum of the infinite series

e = $\frac{1}{1!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1*2} + \frac{1}{1*2*3}+ ...$

Answer 2

Answer/explanation

lim h → 0 (e^h − 1) /h = 1.

e is the number such that lim h → [infinity] e^(h + 1 )/h = 1.

e is the number such that lim h → 0 (e^h + 1)/h = 1.

e is the number such that lim h → 1 (e^−h − 1)/ h = 1

From the limits above, e is defined as a mathematical constant approximately equal to 2.71828.

The value of e is the base of the natural logarithm

(b) As h appoaches infinity, lim h → 0 (2.7^h − 1 )/h = lim h → 0 (2.8^h − 1 )/h = 2.72 to 2 decimal places.

It can be concluded that 2.70 < e < 2.80