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The problem provided asks for the evaluation of a limit, specifically:

\[
\lim_{x \to 0} \frac{\cos(1x) - \cos(10x)}{x^2}
\]

You are given a table with several values of \( x \) and are asked to calculate the function \( f(x) = \frac{\cos(1x) - \cos(10x)}{x^2} \) for each of those values of \( x \) in the table.

Here’s the table of \( x \)-values that need to be evaluated:

\[
\begin{array}{c|c}
x & f(x) = \frac{\cos(1x) - \cos(10x)}{x^2} \\
\hline
0.01 & \\
0.001 & \\
0.0001 & \\
-0.0001 & \\
-0.001 & \\
-0.01 & \\
\end{array}
\]

We can calculate these values to estimate the limit as \( x \to 0 \). I will calculate them now.

Let me work through these calculations.Here are the calculated values for \( f(x) = \frac{\cos(1x) - \cos(10x)}{x^2} \) at the given points:

\[
\begin{array}{c|c}
x & f(x) \\
\hline
0.01 & 49.458351 \\
0.001 & 49.499583 \\
0.0001 & 49.499996 \\
-0.0001 & 49.499996 \\
-0.001 & 49.499583 \\
-0.01 & 49.458351 \\
\end{array}
\]

Based on these values, it appears that as \( x \) approaches 0, \( f(x) \) approaches approximately **49.5**. Therefore, we can estimate that:

\[
\lim_{x \to 0} \frac{\cos(1x) - \cos(10x)}{x^2} \approx 49.5
\]

Would you like further details on this calculation?

Here are 5 related questions to consider:
1. What methods can be used to rigorously prove the value of this limit?
2. How can the Taylor expansion of cosine help in evaluating limits like this?
3. What are the common forms of indeterminate limits, and how do we resolve them?
4. How do small-angle approximations for trigonometric functions influence limits near zero?
5. Can L'Hopital's Rule be used here, and if so, how?

**Tip**: When evaluating limits involving trigonometric functions, expansions or approximations around zero often simplify the analysis.

Estimate the value of the limit, if it exists, by evaluating the function f(x) = (cos(1x) - cos(10x)) / x^2 for different values of x and reporting the result accurate to at least six decimal places.

Explore how to estimate the limit of the function f(x) = (cos(1x) - cos(10x)) / x^2 as x approaches 0, using accurate calculations to six decimal places. This problem involves key calculus concepts such as limits, trigonometric functions, and L'Hopital's Rule.

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