To solve the given limit using L'Hospital's Rule, we must first evaluate if the limit results in an indeterminate form like $\frac{0}{0}$. The limit is:

$\lim_{x \to 0} \frac{\sin(10x)}{\tan(4x)}$

Step 1: Check for indeterminate form

As $x \to 0$:

• $\sin(10x) \to \sin(0) = 0$
• $\tan(4x) \to \tan(0) = 0$

Thus, the limit is of the form $\frac{0}{0}$, so we can apply L'Hospital's Rule, which states that if the limit results in $\frac{0}{0}$, we take the derivative of the numerator and denominator separately.

Step 2: Differentiate the numerator and denominator

• The derivative of $\sin(10x)$ is $10\cos(10x)$.
• The derivative of $\tan(4x)$ is $4\sec^2(4x)$.

Thus, the new limit becomes:

$\lim_{x \to 0} \frac{10\cos(10x)}{4\sec^2(4x)}$

Step 3: Simplify the expression

At $x = 0$:

• $\cos(10x) \to \cos(0) = 1$
• $\sec(4x) = \frac{1}{\cos(4x)}$, and $\cos(4x) \to \cos(0) = 1$, so $\sec^2(4x) \to 1$

Therefore, the limit simplifies to:

$\lim_{x \to 0} \frac{10 \cdot 1}{4 \cdot 1} = \frac{10}{4} = \frac{5}{2}$

Final Answer:

$\lim_{x \to 0} \frac{\sin(10x)}{\tan(4x)} = \frac{5}{2}$

Would you like more details or have any questions about the steps?

Here are five related questions:

1. How is L'Hospital's Rule applied when dealing with indeterminate forms?
2. What are the common indeterminate forms where L'Hospital's Rule can be applied?
3. How do you differentiate trigonometric functions like $\sin(x)$ and $\tan(x)$?
4. What happens if the limit still results in an indeterminate form after applying L'Hospital's Rule once?
5. Can L'Hospital's Rule be applied to limits involving infinity?

Tip: Always check if the original limit results in an indeterminate form before applying L'Hospital's Rule.