We are tasked with finding the limit:

\[
\lim_{n \to \infty} \frac{1}{5 - \frac{1}{2^n}}.
\]

### Step-by-Step Solution:

1. **Analyze the term \(\frac{1}{2^n}\) as \(n \to \infty\):**
   - As \(n\) approaches infinity, \(2^n\) becomes infinitely large. Hence, \(\frac{1}{2^n}\) approaches 0.
   
   \[
   \lim_{n \to \infty} \frac{1}{2^n} = 0.
   \]

2. **Substitute this result into the expression:**
   - The limit now becomes:
   
   \[
   \lim_{n \to \infty} \frac{1}{5 - 0} = \frac{1}{5}.
   \]

Thus, the value of the limit is:

\[
\boxed{\frac{1}{5}}.
\]

Would you like more details on any of these steps? Here are some related questions to explore:

1. What is the behavior of other exponential terms like \(\frac{1}{3^n}\) as \(n \to \infty\)?
2. How do limits behave when functions include exponents and logarithms?
3. Can you compute \(\lim_{n \to \infty} \frac{1}{10 - \frac{1}{3^n}}\)?
4. How would you solve limits involving fractions and radicals?
5. How does L'Hopital's Rule apply to limits that tend towards indeterminate forms?

**Tip:** When dealing with limits involving exponential terms, remember that \(a^n \to \infty\) as \(n \to \infty\) if \(a > 1\).

Discover how to compute the limit of 1 / (5 - 1 / (2^n)) as n approaches infinity with a detailed solution. This problem demonstrates key concepts in limits and exponential decay, and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation