It seems that the text from the image is not easily readable with the standard OCR method. Let me manually inspect and transcribe the visible mathematical content.

From the image, the formula appears to resemble:

$\lim_{{x \to 0}} \frac{{\sin(3x)}}{x} = 3$

This is a well-known limit in calculus. Let me explain the reasoning behind this result.

Explanation:

We are evaluating the limit:

$\lim_{{x \to 0}} \frac{{\sin(3x)}}{x}$

Using the substitution $u = 3x$, we rewrite the limit as:

$\lim_{{x \to 0}} \frac{{\sin(3x)}}{x} = \lim_{{u \to 0}} \frac{{\sin(u)}}{\frac{u}{3}} = 3 \lim_{{u \to 0}} \frac{{\sin(u)}}{u}$

It is a standard result that:

$\lim_{{u \to 0}} \frac{{\sin(u)}}{u} = 1$

Therefore, the limit simplifies to:

$3 \times 1 = 3$

Thus, the solution is:

$\lim_{{x \to 0}} \frac{{\sin(3x)}}{x} = 3$

Would you like more details, or do you have any questions about this solution?

Related Questions:

1. What are the common limit properties of trigonometric functions like sine and cosine?
2. How does L'Hôpital's Rule apply to trigonometric limits?
3. What other substitution methods work for solving trigonometric limits?
4. How can this type of limit be generalized for other angles, such as $\sin(nx)$?
5. Can this limit be solved using series expansions of sine?

Tip: For evaluating limits involving trigonometric functions, always check if you can apply standard limits like $\lim_{{x \to 0}} \frac{{\sin(x)}}{x} = 1$, which simplifies many problems.