https://file.aiquickdraw.com/mathbot/file/172559021766813be9iyo.jpg

The expression in the image is:

\[
\lim_{{x \to 0}} \frac{\sin(3x)}{5x}
\]

To solve this limit, we can use the standard limit:

\[
\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1
\]

Rewriting the expression in a more convenient form, we can introduce a factor of 3 in the numerator and denominator:

\[
\lim_{{x \to 0}} \frac{\sin(3x)}{5x} = \lim_{{x \to 0}} \frac{3 \cdot \sin(3x)}{3 \cdot 5x} = \frac{3}{5} \lim_{{x \to 0}} \frac{\sin(3x)}{3x}
\]

Now, using the standard limit mentioned earlier:

\[
\lim_{{x \to 0}} \frac{\sin(3x)}{3x} = 1
\]

Substitute this result back:

\[
\frac{3}{5} \cdot 1 = \frac{3}{5}
\]

Thus, the limit is:

\[
\lim_{{x \to 0}} \frac{\sin(3x)}{5x} = \frac{3}{5}
\]

Would you like more details or have any questions?

### Further Related Questions:
1. How can we use L'Hopital's Rule to solve this limit?
2. What is the limit of \(\frac{\sin(kx)}{mx}\) as \(x \to 0\) for any constants \(k\) and \(m\)?
3. How do we evaluate limits involving trigonometric functions?
4. What is the proof of the standard limit \(\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1\)?
5. How do we compute limits of the form \(\lim_{{x \to a}} \frac{f(x)}{g(x)}\) where both \(f(a) = 0\) and \(g(a) = 0\)?

### Tip:
When solving limits involving trigonometric functions, consider rewriting them in forms that make use of standard limits like \(\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1\).

Learn how to compute the limit of sin(3x)/5x as x approaches 0. This step-by-step guide uses standard limit theorems and trigonometric principles to find the solution, making it suitable for students in Grades 10-12.

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