The value of the limit lim x -> 0 (tan^2 (7x) - tan^2 (3x))/(tan^2 (5x) - tan^2 x)

To find the value of the limit:

\[
L = \lim_{x \to 0} \frac{\tan^2(7x) - \tan^2(3x)}{\tan^2(5x) - \tan^2(x)}
\]

we start by applying the small-angle approximation for the tangent function. When \(x \to 0\), \(\tan(x) \approx x\). Therefore:

\[
\tan(7x) \approx 7x, \quad \tan(3x) \approx 3x, \quad \tan(5x) \approx 5x, \quad \text{and} \quad \tan(x) \approx x.
\]

Substitute these approximations into the limit:

\[
L \approx \lim_{x \to 0} \frac{(7x)^2 - (3x)^2}{(5x)^2 - x^2}.
\]

Simplify the expression:

\[
L = \lim_{x \to 0} \frac{49x^2 - 9x^2}{25x^2 - x^2} = \lim_{x \to 0} \frac{40x^2}{24x^2}.
\]

This simplifies to:

\[
L = \lim_{x \to 0} \frac{40}{24} = \frac{5}{3}.
\]

Thus, the value of the limit is:

\[
\boxed{\frac{5}{3}}.
\]

Would you like to see the details of the approximation steps, or do you have any other questions? 

Here are 5 related questions you might find interesting:
1. How does the small-angle approximation work for trigonometric functions?
2. What are higher-order approximations for trigonometric functions near zero?
3. How would this limit change if we used different coefficients for \(x\)?
4. What is the significance of Taylor series expansion in solving limits?
5. How do you find limits involving trigonometric functions without using approximations?

**Tip:** For limits involving trigonometric functions, always check if small-angle approximations can simplify the expression.

Learn how to evaluate the limit lim x -> 0 (tan^2(7x) - tan^2(3x))/(tan^2(5x) - tan^2(x)) using trigonometric identities and small-angle approximation. This problem explores trigonometry and limit evaluation, suitable for advanced high school students and beyond.

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