Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)
lim
x→0
x6 cos(7/x)

We are tasked with evaluating the following limit:

\[
\lim_{x \to 0} x^6 \cos\left(\frac{7}{x}\right)
\]

### Step-by-step solution:

1. **Behavior of the cosine function**:  
   The function \(\cos\left(\frac{7}{x}\right)\) oscillates between -1 and 1 for all values of \(x \neq 0\), since cosine is a bounded function. Therefore, we can write:

   \[
   -1 \leq \cos\left(\frac{7}{x}\right) \leq 1
   \]

2. **Multiplying by \(x^6\)**:  
   Now, multiply both sides of the inequality by \(x^6\):

   \[
   -x^6 \leq x^6 \cos\left(\frac{7}{x}\right) \leq x^6
   \]

3. **Taking the limit**:  
   As \(x \to 0\), \(x^6 \to 0\). This applies to both the upper and lower bounds of the inequality:

   \[
   \lim_{x \to 0} -x^6 = 0 \quad \text{and} \quad \lim_{x \to 0} x^6 = 0
   \]

   Therefore, by the **squeeze theorem**:

   \[
   \lim_{x \to 0} x^6 \cos\left(\frac{7}{x}\right) = 0
   \]

### Final Answer:
\[
\boxed{0}
\]

Would you like more details or have any questions? Here are five related questions for further exploration:

1. How does the squeeze theorem work, and in which situations can it be applied?
2. Can we use the same approach to find limits for functions like \(x^n \sin\left(\frac{a}{x}\right)\) as \(x \to 0\)?
3. Why does cosine oscillate but not affect the limit result in this case?
4. What would happen if the exponent of \(x\) was less than 1, say \(x^2\), instead of \(x^6\)?
5. How would this limit change if it involved a function like \(\sin\left(\frac{7}{x}\right)\) instead of cosine?

**Tip**: The squeeze theorem is particularly useful for functions with oscillatory terms like sine or cosine when they are bounded and multiplied by a term approaching zero.

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim x→0 x^6 cos(7/x)

Explore the limit problem lim x→0 x^6 cos(7/x) and learn how to apply the Squeeze Theorem. This step-by-step solution discusses the behavior of oscillatory functions like cosine, how bounds are used, and why the limit evaluates to 0. Ideal for college-level calculus students.

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