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:

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

### Step-by-step solution:

1. **Examine the behavior of the cosine function**: 
   \[
   \cos(7x) 
   \]
   The cosine function is continuous everywhere, and specifically, at \( x = 0 \), we have:
   \[
   \cos(7 \cdot 0) = \cos(0) = 1
   \]

2. **Simplifying the expression**:
   We now rewrite the limit, knowing that \( \cos(7x) \) approaches 1 as \( x \to 0 \):
   \[
   \lim_{x \to 0} x^6 \cos(7x) = \lim_{x \to 0} x^6 \cdot 1 = \lim_{x \to 0} x^6
   \]

3. **Evaluate the limit of \( x^6 \) as \( x \to 0 \)**:
   The function \( x^6 \) approaches 0 as \( x \) approaches 0:
   \[
   \lim_{x \to 0} x^6 = 0
   \]

Therefore, the limit is:

\[
\boxed{0}
\]

Would you like any more details or clarifications?

---

Here are 5 related questions you might find interesting:
1. How do you find the limit of a product involving trigonometric functions?
2. What is the limit of \( x^2 \sin(x) \) as \( x \to 0 \)?
3. How does L'Hôpital's Rule apply to limits involving indeterminate forms?
4. What happens when you take the limit of higher powers of \( x \), like \( x^8 \), as \( x \to 0 \)?
5. Can you explain how continuity helps in simplifying limits?

**Tip**: When dealing with limits, always check if the trigonometric functions can be simplified near 0 by using known values or approximations (like \( \cos(0) = 1 \) or \( \sin(0) = 0 \)).

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

Learn how to evaluate the limit of x^6 cos(7x) as x approaches 0 with a step-by-step solution. This problem involves key calculus concepts such as limits and trigonometric functions, suitable for students in Grades 11-12.

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