Use the epsilon delta theory of limit to prove lim x->1 1/x = 1

To prove that \(\lim_{x \to 1} \frac{1}{x} = 1\) using the epsilon-delta definition of a limit, we start with the formal definition:

For every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 1| < \delta\), then \(|\frac{1}{x} - 1| < \epsilon\).

### Step 1: Express the Limit Condition

We want to show:

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

This simplifies to:

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

### Step 2: Manipulate the Inequality

Rearranging gives:

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

This implies:

\[
|1 - x| < \epsilon |x|
\]

### Step 3: Choose \(\delta\)

We want to find a suitable \(\delta\) such that if \(0 < |x - 1| < \delta\), then \(x\) is close to 1. We can bound \(|x|\):

If \(x\) is close to 1, specifically, if \(|x - 1| < \frac{1}{2}\), then:

\[
\frac{1}{2} < x < \frac{3}{2}
\]

Thus, \(|x|\) is bounded away from zero. In this case, we have:

\[
\frac{1}{2} < |x| < \frac{3}{2}
\]

### Step 4: Finalizing \(\epsilon\)

Now, since \(|x| \geq \frac{1}{2}\), we can replace \(|x|\) in our inequality:

\[
|1 - x| < \epsilon |x| \implies |1 - x| < 2\epsilon
\]

### Step 5: Set \(\delta\)

Let \(\delta = \min\left(\frac{1}{2}, 2\epsilon\right)\). Therefore, if \(0 < |x - 1| < \delta\), then:

\[
|1 - x| < 2\epsilon \implies \left| \frac{1}{x} - 1 \right| < \epsilon
\]

### Conclusion

We have shown that for every \(\epsilon > 0\), we can find a \(\delta > 0\) such that:

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

### Questions to Consider
1. How does the choice of \(\delta\) affect the proof?
2. Can this method be used for other limits?
3. What would change if we proved a limit as \(x \to 0\)?
4. How does the concept of continuity relate to limits?
5. What are common pitfalls in using epsilon-delta definitions?

**Tip:** Always visualize the limit on a graph to understand the behavior of the function near the limit point. Would you like more details on any part of this proof?

Explore the epsilon-delta proof demonstrating that lim x->1 1/x = 1. This comprehensive guide covers the formal definition of limits, key steps in the proof, and related mathematical concepts, making it ideal for advanced high school students.

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