https://file.aiquickdraw.com/mathbot/file/1726524845461j2rndmwy.jpg

I see the image contains three limit problems: 11, 13, and 15. Let's solve each one step by step.

### Problem 11:
\[
\lim_{{x \to 0^+}} \frac{{\sin(4x)}}{{5x}}
\]
Here, we can use the limit property: \(\lim_{x \to 0} \frac{\sin(kx)}{x} = k\), which holds true because \(\frac{\sin(4x)}{x}\) approaches \(4\) as \(x \to 0\).

Thus:
\[
\lim_{{x \to 0^+}} \frac{{\sin(4x)}}{{5x}} = \frac{4}{5}
\]

### Problem 13:
\[
\lim_{{x \to \infty}} \frac{{2x^3 - 7x^2}}{{4x^3 + x + 9}}
\]
When taking the limit as \(x \to \infty\), we divide both the numerator and denominator by the highest power of \(x\), which is \(x^3\):
\[
\lim_{{x \to \infty}} \frac{{2x^3 - 7x^2}}{{4x^3 + x + 9}} = \lim_{{x \to \infty}} \frac{{2 - \frac{7}{x}}}{{4 + \frac{1}{x^2} + \frac{9}{x^3}}}
\]
As \(x\) approaches infinity, the terms with \(x\) in the denominator will approach zero:
\[
= \frac{2}{4} = \frac{1}{2}
\]

### Problem 15:
\[
\lim_{{x \to 3}} f(x)
\]
where
\[
f(x) =
\begin{cases}
9 - x^2, & x < 3 \\
2x - 3, & x \geq 3
\end{cases}
\]
We need to evaluate the left-hand limit (\(x \to 3^-\)) and the right-hand limit (\(x \to 3^+\)):

- For \(x \to 3^-\), the function is \(f(x) = 9 - x^2\), so \(\lim_{{x \to 3^-}} f(x) = 9 - 3^2 = 0\).
- For \(x \to 3^+\), the function is \(f(x) = 2x - 3\), so \(\lim_{{x \to 3^+}} f(x) = 2(3) - 3 = 3\).

Since the left-hand limit (\(0\)) does not equal the right-hand limit (\(3\)), the limit does not exist.

### Summary of Answers:
1. \(\lim_{{x \to 0^+}} \frac{{\sin(4x)}}{{5x}} = \frac{4}{5}\)
2. \(\lim_{{x \to \infty}} \frac{{2x^3 - 7x^2}}{{4x^3 + x + 9}} = \frac{1}{2}\)
3. \(\lim_{{x \to 3}} f(x)\) does not exist.

Let me know if you'd like any additional explanations or details! Here are five related questions to expand on this:

1. How do we apply L'Hopital's Rule to limits involving indeterminate forms?
2. Why do higher-degree terms dominate in limits as \(x \to \infty\)?
3. What are the conditions for the existence of a limit for a piecewise function?
4. How can trigonometric limits be evaluated using standard limit properties?
5. When does a limit approaching infinity result in 0?

**Tip**: Always check for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), which may require L'Hopital's Rule or simplification techniques.

Solve the limit problems: lim_{x → 0^+} (sin(4x) / 5x), lim_{x → ∞} ((2x^3 - 7x^2) / (4x^3 + x + 9)), lim_{x → 3} f(x) where f(x) is a piecewise function.

This guide provides step-by-step solutions to limit problems including trigonometric limits, polynomial limits at infinity, and limits involving piecewise functions. These problems are ideal for students in Grades 11-12 studying calculus.

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