https://file.aiquickdraw.com/mathbot/file/1726181866739xjvfxvl1.jpg

You have provided a scenario involving a formal definition of limits as \(x \to \infty\), specifically for a constant function \( f(x) = k \). To complete the statement using the definition of the limit, we proceed as follows:

### Complete Introductory Statement
The task is to use the formal definition of limits to establish that for a constant function \( f(x) = k \), the limit as \( x \) approaches infinity is \( k \). Here is how you can formally write it:

Given any \( \epsilon > 0 \), we need to find a number \( M \) such that for all \( x \), \( |f(x) - k| < \epsilon \) whenever \( x > M \). For a constant function \( f(x) = k \), this simplifies to finding \( M \) such that \( |k - k| < \epsilon \) for \( x > M \).

### Simplification
Since \( |k - k| = 0 \), which is always less than any positive \( \epsilon \), the condition \( |k - k| < \epsilon \) is trivially satisfied regardless of the value of \( x \).

### Conclusion
Hence, you can choose any \( M \), say \( M = 1 \) (or any other real number), and the condition \( |f(x) - k| < \epsilon \) whenever \( x > M \) will be satisfied for any \( \epsilon > 0 \).

This concludes that \( \lim_{x \to \infty} f(x) = k \).

Do you need further details on this solution or have any questions? Here are some follow-up topics you might consider:
1. How would you apply a similar proof for a limit as \( x \to -\infty \)?
2. What changes if the function \( f(x) \) is not constant?
3. How does the concept of limit apply in the context of sequences?
4. Could this formal definition be used to prove limits involving more complex functions?
5. How does the formal definition relate to the concept of continuity?

**Tip:** When dealing with limits and continuous functions, always consider the behavior of the function over its entire domain and how small changes in the input (within a specified range) affect the output. This helps in understanding and proving limits more effectively.

Use formal definitions of limits as x → ± ∞ to establish the limit. If f has the constant value f(x) = k, then lim f(x) = k as x → ∞. Complete the introductory statement. Let ε > 0 be given. Find a number M such that for all x, |k - k| = □ < ε whenever x > M.

Explore how to establish the limit of a constant function using the epsilon-delta definition of limits. Learn to find M for any ε > 0 such that |f(x) - k| < ε for all x > M when f(x) = k. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation