Use the ϵ, δ-definition of the limit of a function to prove
lim x→0 x/(1 − x2) = 0.

To prove the limit \(\lim_{x \to 0} \frac{x}{1 - x^2} = 0\) using the \(\epsilon, \delta\)-definition of a limit, we need to show that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - 0| < \delta\), it follows that \(\left|\frac{x}{1 - x^2} - 0\right| < \epsilon\).

### Step-by-Step Proof

1. **Function Simplification**:
   We are given the function \(f(x) = \frac{x}{1 - x^2}\), and we need to prove:
   \[
   \lim_{x \to 0} \frac{x}{1 - x^2} = 0.
   \]
   This means for any \(\epsilon > 0\), we need to find a \(\delta > 0\) such that:
   \[
   0 < |x| < \delta \implies \left| \frac{x}{1 - x^2} \right| < \epsilon.
   \]

2. **Inequality for \(\frac{x}{1 - x^2}\)**:
   We start by manipulating the expression \(\left| \frac{x}{1 - x^2} \right|\):
   \[
   \left| \frac{x}{1 - x^2} \right| = \frac{|x|}{|1 - x^2|}.
   \]
   Notice that as \(x \to 0\), \(1 - x^2\) approaches 1. To control this term, we observe that for \(x\) close to 0, \(|1 - x^2|\) remains bounded away from 0. Specifically, for small values of \(x\), we have \(|1 - x^2| \geq \frac{1}{2}\) since \(x^2\) is small.

3. **Bounding \(\frac{1}{1 - x^2}\)**:
   Choose \(\delta \leq 1\), so that for \(x \in (-\delta, \delta)\), we have:
   \[
   |x| < 1 \implies 1 - x^2 > \frac{1}{2}.
   \]
   This gives:
   \[
   \frac{1}{|1 - x^2|} \leq 2.
   \]
   Therefore:
   \[
   \left| \frac{x}{1 - x^2} \right| \leq 2|x|.
   \]

4. **Choosing \(\delta\)**:
   Now, to ensure that \(\left| \frac{x}{1 - x^2} \right| < \epsilon\), we want:
   \[
   2|x| < \epsilon.
   \]
   Solving for \(|x|\), we get:
   \[
   |x| < \frac{\epsilon}{2}.
   \]
   Thus, we can choose \(\delta = \min\left(1, \frac{\epsilon}{2}\right)\).

5. **Conclusion**:
   Therefore, for any \(\epsilon > 0\), if we choose \(\delta = \min\left(1, \frac{\epsilon}{2}\right)\), then whenever \(0 < |x| < \delta\), we have:
   \[
   \left| \frac{x}{1 - x^2} \right| < \epsilon.
   \]
   This proves that \(\lim_{x \to 0} \frac{x}{1 - x^2} = 0\) using the \(\epsilon, \delta\)-definition of the limit.

---

Would you like more details on any specific part of the proof, or do you have any other questions?

Here are 5 related questions to explore:

1. How does the \(\epsilon, \delta\)-definition apply to limits involving infinity?
2. Can you apply the \(\epsilon, \delta\)-definition to prove \(\lim_{x \to 1} \frac{1}{x} = 1\)?
3. What are some common mistakes when using the \(\epsilon, \delta\)-definition of limits?
4. How can we extend the \(\epsilon, \delta\)-definition to limits of multivariable functions?
5. How does the choice of \(\delta\) change if the function has vertical asymptotes near the limit?

**Tip**: When applying the \(\epsilon, \delta\)-definition, always focus on bounding the troublesome part of the function to control its behavior near the limit point.

Use the ϵ, δ-definition of the limit of a function to prove lim x→0 x/(1 − x^2) = 0.

In this problem, we prove the limit lim x→0 x/(1 − x^2) = 0 using the epsilon-delta definition of a limit. The solution includes a step-by-step breakdown of how to handle rational functions near 0 and how to choose appropriate delta values for any given epsilon. Suitable for university-level Calculus I students.

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