Prove the precise definition for each limit statement lim(1/x)=1/6

To prove the precise definition of the limit statement:

\[
\lim_{{x \to 6}} \frac{1}{x} = \frac{1}{6}
\]

we need to use the **ε-δ definition** of a limit. According to this definition:

For every \( \epsilon > 0 \), there must exist a \( \delta > 0 \) such that for all \( x \) satisfying \( 0 < |x - 6| < \delta \), it follows that \( \left| \frac{1}{x} - \frac{1}{6} \right| < \epsilon \).

### Step-by-Step Proof:

1. **Start with the inequality** \( \left| \frac{1}{x} - \frac{1}{6} \right| < \epsilon \).

We want to manipulate this inequality to express \( |x - 6| \) in terms of \( \epsilon \).

\[
\left| \frac{1}{x} - \frac{1}{6} \right| = \left| \frac{6 - x}{6x} \right|
\]

This simplifies to:

\[
\left| \frac{6 - x}{6x} \right| < \epsilon
\]

2. **Isolate \( |6 - x| \):**

Multiply both sides by \( 6x \):

\[
|6 - x| < 6x\epsilon
\]

3. **Find bounds on \( x \):**

Next, we place some bounds on \( x \) so that \( 6x \) stays controlled. Since we're considering \( x \) near 6, let's assume \( |x - 6| < 1 \), which implies:

\[
5 < x < 7
\]

For \( x \) in this interval, the value of \( 6x \) satisfies:

\[
30 < 6x < 42
\]

Thus, we can say that \( 6x \) is bounded below by 30. This gives us:

\[
|6 - x| < 6x\epsilon \quad \Rightarrow \quad |6 - x| < 30\epsilon
\]

4. **Choose \( \delta \):**

To ensure that \( \left| \frac{1}{x} - \frac{1}{6} \right| < \epsilon \), we must choose \( \delta \) so that \( |x - 6| < \delta \) implies \( |x - 6| < 30\epsilon \). Thus, we set:

\[
\delta = 30\epsilon
\]

### Conclusion:

For any \( \epsilon > 0 \), if we choose \( \delta = 30\epsilon \), then for all \( x \) such that \( 0 < |x - 6| < \delta \), it follows that:

\[
\left| \frac{1}{x} - \frac{1}{6} \right| < \epsilon
\]

This completes the proof using the ε-δ definition of a limit.

Would you like further details on any step, or perhaps some additional examples? Here are some related questions for further exploration:

1. How does the ε-δ definition of a limit work for functions like \( \frac{1}{x} \)?
2. What is the geometric interpretation of the ε-δ definition of a limit?
3. How do you handle limits involving infinity with the ε-δ definition?
4. How can the δ value change based on different functions?
5. What happens if you pick different bounds for \( x \) in the proof?

**Tip:** When working with limits, simplifying the expression within the limit is often key to identifying how \( \delta \) depends on \( \epsilon \).

Explore the detailed proof of the limit statement lim(1/x) = 1/6 using the ε-δ definition. This step-by-step explanation includes key calculus concepts, formulas, and theorems. Suitable for Calculus 1 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