To prove that

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

we can use the Squeeze Theorem (also known as the Sandwich Theorem).

Step-by-Step Proof

1. Understand the Function Behavior:

   The sine function, $\sin\left(\frac{1}{x}\right)$, oscillates between $-1$ and $1$ for all values of $x$, so we have:

   $-1 \leq \sin\left(\frac{1}{x}\right) \leq 1.$

2. Multiply by $x$:

   Multiply the entire inequality by $x$. Since $x$ can be positive or negative, we'll consider two cases:

   • For $x > 0$:

     $-x \leq x \sin\left(\frac{1}{x}\right) \leq x.$

   • For $x < 0$:

     $x \leq x \sin\left(\frac{1}{x}\right) \leq -x.$

   Notice that these inequalities hold for $x$ approaching 0 from either side.

3. Consider the Limit of the Bounds:

   As $x \to 0$, both $x$ and $-x$ approach 0. Therefore:

   $\lim_{x \to 0^+} x = 0 \quad \text{and} \quad \lim_{x \to 0^+} -x = 0.$

   Similarly,

   $\lim_{x \to 0^-} x = 0 \quad \text{and} \quad \lim_{x \to 0^-} -x = 0.$

   Hence, in either case:

   $\lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0.$

4. Apply the Squeeze Theorem:

   Since $x \sin\left(\frac{1}{x}\right)$ is squeezed between two functions that both tend to 0 as $x \to 0$, by the Squeeze Theorem:

   $\lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0.$

This completes the proof.

Would you like any further details or explanations on this proof?

Here are 5 related questions to expand your understanding:

1. How does the Squeeze Theorem work in different types of limits?
2. Can you prove the limit using an epsilon-delta definition?
3. What happens to the function $x \sin\left(\frac{1}{x}\right)$ as $x$ approaches infinity?
4. How does the limit change if we replace $x \sin\left(\frac{1}{x}\right)$ with $x^2 \sin\left(\frac{1}{x}\right)$?
5. What are some other common functions that require the Squeeze Theorem for limit proofs?

Tip: When working with limits involving oscillating functions, the Squeeze Theorem is a powerful tool, especially if you can bound the function between two simpler functions that have known limits.