To prove using the ε-δ (epsilon-delta) definition of a limit that:

$\lim_{x \to 0} x \cos \left( \frac{1}{x} \right) = 0,$

we must demonstrate that for every ε > 0, there exists a corresponding δ > 0 such that whenever $0 < |x| < \delta$, the following inequality holds:

$\left| x \cos \left( \frac{1}{x} \right) - 0 \right| = \left| x \cos \left( \frac{1}{x} \right) \right| < \epsilon.$

Step-by-step proof:

1. Starting with the absolute value of the function:

   We want to prove that: $\left| x \cos \left( \frac{1}{x} \right) \right| < \epsilon.$

2. Bound the cosine function:

   Since $\cos \left( \frac{1}{x} \right)$ is bounded for all $x \neq 0$ (because cosine always satisfies $-1 \leq \cos(\theta) \leq 1$ for any real $\theta$), we have: $-1 \leq \cos \left( \frac{1}{x} \right) \leq 1.$

   This implies: $\left| \cos \left( \frac{1}{x} \right) \right| \leq 1.$

3. Simplifying the expression:

   Using the bound $\left| \cos \left( \frac{1}{x} \right) \right| \leq 1$, we get: $\left| x \cos \left( \frac{1}{x} \right) \right| \leq |x|.$

4. Applying the ε-δ definition:

   Now we need to make sure that for any ε > 0, we can find a δ > 0 such that: $|x| < \delta \implies |x| < \epsilon.$

   This choice of δ is straightforward: we can simply choose $\delta = \epsilon$. Therefore, if $|x| < \delta$, we have: $|x \cos \left( \frac{1}{x} \right)| \leq |x| < \epsilon.$

5. Conclusion:

   Thus, by choosing $\delta = \epsilon$, we satisfy the condition required by the ε-δ definition. Therefore, we have proven that: $\lim_{x \to 0} x \cos \left( \frac{1}{x} \right) = 0.$

This completes the proof using the epsilon-delta definition of a limit.

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Would you like to go deeper into any part of the explanation or have any questions?

Here are some related questions:

1. How does the bounded nature of cosine help in epsilon-delta proofs?
2. Can we generalize this technique for other oscillatory functions?
3. How do we choose δ in more complex limits?
4. What happens if the cosine function was unbounded near 0?
5. Why is epsilon-delta definition crucial in proving limits formally?

Tip: When dealing with oscillating functions like $\cos(1/x)$, bounding them is a useful strategy in epsilon-delta proofs.