Let's solve the given limit using L'Hopital's Rule. We are asked to compute:

$\lim_{x \to 0^+} \frac{4x}{\tan(5x)}$

Step 1: Check if we can apply L'Hopital's Rule

L'Hopital's Rule is applicable if the limit results in an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

• As $x \to 0^+$, the numerator $4x \to 0$.
• The denominator $\tan(5x) \to \tan(0) = 0$.

Thus, we have the indeterminate form $\frac{0}{0}$, so we can apply L'Hopital's Rule.

Step 2: Differentiate the numerator and denominator

Using L'Hopital's Rule, we differentiate the numerator and denominator with respect to $x$:

• Derivative of the numerator $4x$ is $4$.
• Derivative of the denominator $\tan(5x)$ is $5 \sec^2(5x)$, by applying the chain rule.

Step 3: Apply the limit again

Now we need to compute:

$\lim_{x \to 0^+} \frac{4}{5 \sec^2(5x)}$

Since $\sec(5x) = \frac{1}{\cos(5x)}$, we know that as $x \to 0^+$, $\cos(5x) \to 1$, so $\sec^2(5x) \to 1^2 = 1$.

Thus, the limit becomes:

$\frac{4}{5 \cdot 1} = \frac{4}{5}$

Final Answer:

The value of the limit is $\frac{4}{5}$.

Would you like further clarification on any of these steps?

Related Questions:

1. How do you apply L'Hopital's Rule to more complicated indeterminate forms like $\frac{\infty}{\infty}$?
2. What are the conditions under which L'Hopital's Rule can be used?
3. How does the chain rule apply in differentiation, as seen in this problem with $\tan(5x)$?
4. How can we deal with limits that lead to forms like $\frac{0}{\infty}$?
5. What alternative methods can be used to solve limits when L'Hopital's Rule is not applicable?

Tip:

Always verify that you have an indeterminate form before applying L'Hopital's Rule, as it is only valid under certain conditions.