Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.)

 Let f(x)= (cos(5x)-cos(7x))/(x^(2)) 

We want to find the limit lim x is 0 f(x)= lim x is 0 (cos(5x)-cos(7x))/(x^(2)) 

Start by calculating the values of the function for the inputs listed in this table. 

x 
0.2 
0.1 
0.05 
0.01 
0.001 
0.0001 
0.00001 

Find the f(x) inputs in the full decimal answer 

Based on the values in this table, it appears 

lim x is 0 (cos(5x)-cos(7x))/(x^(2)) =

Here are the calculated values of \( f(x) = \frac{\cos(5x) - \cos(7x)}{x^2} \) for the given inputs:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0.2 & 9.258379 \\
0.1 & 11.274037 \\
0.05 & 11.815884 \\
0.01 & 11.992601 \\
0.001 & 11.999926 \\
0.0001 & 11.999999 \\
0.00001 & 11.999999 \\
\hline
\end{array}
\]

Based on the pattern of these values, it appears that:

\[
\lim_{x \to 0} \frac{\cos(5x) - \cos(7x)}{x^2} = 12
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. Why does the limit of this function exist as \(x \to 0\)?
2. What would happen if we used higher precision for \(x\)?
3. How is L'Hôpital's Rule applied to confirm this limit?
4. What happens to the behavior of \(\cos(kx)\) as \(x\) approaches 0 for different values of \(k\)?
5. How does the second derivative of the cosine function relate to this limit?

**Tip**: When functions involve trigonometric expressions, it can be useful to apply Taylor expansions to analyze their behavior near certain points.

Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x)= (cos(5x)-cos(7x))/(x^(2)) We want to find the limit lim x → 0 f(x) = lim x → 0 (cos(5x)-cos(7x))/(x^(2)) Start by calculating the values of the function for the inputs listed in this table: 0.2, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001. Based on the values in this table, it appears lim x → 0 (cos(5x)-cos(7x))/(x^(2)) =

Discover how to find the limit of f(x) = (cos(5x) - cos(7x)) / x^2 as x approaches 0. Using trigonometric and calculus techniques, this problem involves applying L'Hôpital's Rule and Taylor series expansions. Explore the function values for specific inputs, and learn how to evaluate the limit to six decimal places.

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