Can’t you find it other way than h’lopital

Certainly! We can solve this limit problem without using L'Hôpital's Rule by leveraging the Taylor series expansion for \(\sin(x)\) or by simplifying the expression using algebraic techniques.

### Step 1: Use the Taylor Series Expansion
The Taylor series expansion of \(\sin(x)\) around \( x = 0 \) is:

\[
\sin(x) = x - \frac{x^3}{6} + \text{higher-order terms}
\]

For small values of \(x\), we can approximate \(\sin(x)\) by its first term:

\[
\sin(x) \approx x
\]

### Step 2: Simplify the Expression
Substitute this approximation into the limit:

\[
\lim_{{x \to 0}} \frac{5x^2 + 9x}{\sin(x)} \approx \lim_{{x \to 0}} \frac{5x^2 + 9x}{x}
\]

Now, factor the numerator:

\[
\lim_{{x \to 0}} \frac{x(5x + 9)}{x}
\]

### Step 3: Simplify Further
Cancel the \(x\) terms in the numerator and denominator (note that \(x \neq 0\) for this cancellation):

\[
\lim_{{x \to 0}} (5x + 9)
\]

### Step 4: Evaluate the Limit
Now, directly substitute \(x = 0\) into the simplified expression:

\[
5(0) + 9 = 9
\]

Thus, the limit is:

\[
\lim_{{x \to 0}} \frac{5x^2 + 9x}{\sin(x)} = 9
\]

This method avoids the use of L'Hôpital's Rule by using the approximation for \(\sin(x)\) and straightforward algebra.

Do you have any more questions or would you like to explore other approaches?

Here are five related questions for further exploration:
1. How does the Taylor series for \(\sin(x)\) help in finding limits?
2. Can you generalize the process for finding limits involving small-angle approximations?
3. What would the limit be if the numerator were \(x^3\) instead of \(5x^2 + 9x\)?
4. How can we justify the cancellation of \(x\) in the numerator and denominator?
5. How does this approach differ from using L'Hôpital's Rule in terms of understanding the behavior of the function?

**Tip:** Small-angle approximations like \(\sin(x) \approx x\) are extremely useful for solving limits, especially when \(x\) approaches zero.

Learn how to solve a limit problem without using L'Hôpital's Rule by leveraging the Taylor series expansion for sin(x). This detailed solution covers algebraic techniques and small-angle approximations, suitable for advanced high school students.

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