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.

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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.