Question

To determine the value of \(\mathop {\lim }\limits_{x \to \infty } {(1.001)^x}\).

Short answer

The value of \(\mathop {\lim }\limits_{x \to \infty } {(1.001)^x}\) is \(\infty \).

Step-by-step solution

Given information

The expression is \(\mathop {\lim }\limits_{x \to \infty } {(1.001)^x}\).

Step 2: Concept of \(a > 1\)

If\(a > 1\), then\(\mathop {\lim }\limits_{x \to \infty } {(a)^x} \to \infty \)and\(\mathop {\lim }\limits_{x \to - \infty } {(a)^x} = 0\).

Obtain the value of \(\mathop {\lim }\limits_{x \to \infty } {(1.001)^x}\)

From the given, the limit is observed to be in the form of\(\mathop {\lim }\limits_{x \to \infty } {(a)^x}\), where\(a\)is\(1.001\).

It is also noticed that the value of\(a\)is greater than\(1\).

Then by the result,\(\mathop {\lim }\limits_{x \to \infty } {(1.001)^x} = \infty \).

Therefore, the value of \(\mathop {\lim }\limits_{x \to \infty } {(1.001)^x}\) is \(\infty \).