The problem you have provided is:

$\lim_{{x \to \frac{\pi}{2}}} \left( x - \frac{\pi}{2} \right) \tan(3x)$

Solution:

We can approach this problem by analyzing the limit. The expression involves the tangent function, which has vertical asymptotes at odd multiples of $\frac{\pi}{2}$, including $\frac{\pi}{2}$.

Let's break it down step by step:

1. As $x \to \frac{\pi}{2}$, $\tan(3x)$ will tend towards infinity or negative infinity because $3x$ approaches $\frac{3\pi}{2}$, and $\tan\left(\frac{3\pi}{2}\right)$ has a vertical asymptote at this point. Therefore, $\tan(3x)$ becomes very large near this limit.

2. We are multiplying $\tan(3x)$ by $\left( x - \frac{\pi}{2} \right)$, which tends to zero as $x$ approaches $\frac{\pi}{2}$.

We have an indeterminate form $0 \times \infty$, which means we should try using L'Hopital's Rule or equivalent methods to resolve this limit.

Applying L'Hopital's Rule:

We rewrite the limit in a form that allows us to apply L'Hopital’s Rule. Notice:

$\lim_{{x \to \frac{\pi}{2}}} \left( x - \frac{\pi}{2} \right) \tan(3x) = \lim_{{x \to \frac{\pi}{2}}} \frac{x - \frac{\pi}{2}}{\cot(3x)}$

This is now a $\frac{0}{0}$ form, so we can apply L'Hopital's Rule. Differentiating the numerator and denominator:

• The derivative of the numerator $\left( x - \frac{\pi}{2} \right)$ is $1$.
• The derivative of $\cot(3x)$ is $-3\csc^2(3x)$.

Thus, the limit becomes:

$\lim_{{x \to \frac{\pi}{2}}} \frac{1}{-3 \csc^2(3x)}$

As $x \to \frac{\pi}{2}$, $3x \to \frac{3\pi}{2}$, and $\csc\left(\frac{3\pi}{2}\right)$ is $-1$. Hence:

$\lim_{{x \to \frac{\pi}{2}}} \frac{1}{-3 \times (-1)^2} = \frac{1}{-3}$

Final Answer:

The value of the limit is $-\frac{1}{3}$.

Would you like more details on any part of the solution? Here are some related questions for further exploration:

1. What happens to $\tan(x)$ as $x$ approaches $\frac{\pi}{2}$ in general?
2. How does L'Hopital's Rule work, and when is it applicable?
3. Can you use series expansion to solve this limit problem as well?
4. Why is the tangent function undefined at odd multiples of $\frac{\pi}{2}$?
5. What are alternative methods for solving indeterminate forms like $0 \times \infty$?

Tip: When applying L'Hopital's Rule, always check for indeterminate forms like $0/0$ or $\infty/\infty$ before proceeding!