Use epsilon delta theory for limit to prove that lim x->0 (x*cos(1/x))=0

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.

---

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.

This detailed proof shows how to use epsilon-delta theory to prove that the limit of x*cos(1/x) as x approaches 0 is 0. Explore key mathematical concepts such as limits, oscillatory functions, and the epsilon-delta definition, suitable for undergraduate calculus students.

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