lim x->0  sin 7x + tan 3x - sin 5x/tan 9x - tan 3x- sin x soal itu
limit  trigonometri tanpa substitusi

To solve the limit \(\lim_{x \to 0} \frac{\sin(7x) + \tan(3x) - \sin(5x)}{\tan(9x) - \tan(3x) - \sin(x)}\) without substitution, we will use the fact that for small values of \(x\), \(\sin(x) \approx x\) and \(\tan(x) \approx x\). This allows us to approximate each trigonometric function by its linear approximation near 0.

Let's approximate the functions:

\[
\sin(7x) \approx 7x, \quad \sin(5x) \approx 5x, \quad \sin(x) \approx x
\]
\[
\tan(3x) \approx 3x, \quad \tan(9x) \approx 9x
\]

Substituting these approximations into the limit:

\[
\lim_{x \to 0} \frac{7x + 3x - 5x}{9x - 3x - x}
\]

Simplifying the expressions:

\[
\lim_{x \to 0} \frac{(7x + 3x - 5x)}{(9x - 3x - x)} = \lim_{x \to 0} \frac{5x}{5x} = \lim_{x \to 0} 1 = 1
\]

Thus, the limit is \( \boxed{1} \).

Would you like more details or have any questions about this solution? Here are some related questions for further exploration:

1. What is the limit of \(\lim_{x \to 0} \frac{\sin(2x) - \sin(x)}{\tan(4x) - \tan(2x)}\)?
2. How do small angle approximations help in evaluating trigonometric limits?
3. Can the same technique be used for limits involving \(\cos(x)\)?
4. How does the Taylor series expansion of trigonometric functions relate to this limit?
5. What happens if higher-order approximations are used for \(\sin(x)\) and \(\tan(x)\)?

**Tip:** When working with limits of trigonometric functions as \(x\) approaches 0, small angle approximations (\(\sin(x) \approx x\) and \(\tan(x) \approx x\)) are often very useful for simplifying the expression.

Learn how to evaluate the limit \(\lim_{x \to 0} \frac{\sin(7x) + \tan(3x) - \sin(5x)}{\tan(9x) - \tan(3x) - \sin(x)}\) without substitution. Understand the use of small angle approximations in trigonometric limits. Suitable for advanced high school students and college-level calculus learners.

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