Question

select the answers that best complete the given statement. the value that ( left(1+\frac{1}{n}
ight)^{n} ) approaches as ( n ) gets larger and larger is the irrational number called the base. this irrational number is approximately equal to. (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall the limit definition

The limit \(\lim_{n
ightarrow\infty}(1 + \frac{1}{n})^{n}\) is a fundamental concept in calculus.

Step2: Identify the irrational number

The value of \(\lim_{n
ightarrow\infty}(1+\frac{1}{n})^{n}=e\). The number \(e\) is called the natural base.

Step3: Find the approximate value

The approximate value of \(e\) rounded to two decimal places is \(2.71828\approx2.72\)

Answer:

The first blank: \(e\), the second blank: natural, the third blank: \(2.72\)