The image shows a mathematical expression involving a limit and trigonometric functions. The expression is:

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

It seems to be asking for the limit of this expression as $x$ approaches $a$. First, we can cancel out the common factor $(x - a)$ in the numerator and denominator, which simplifies the expression to:

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

Now, substituting $x = a$ into this simplified expression:

$\tan\left(\frac{\pi a}{2a}\right) = \tan\left(\frac{\pi}{2}\right)$

Since $\tan\left(\frac{\pi}{2}\right)$ is undefined (as the tangent function approaches infinity at $\frac{\pi}{2}$), the limit does not exist.

Would you like a further breakdown or have any questions?

Here are 5 related questions:

1. What happens to trigonometric functions when their argument approaches singularities (like $\frac{\pi}{2}$)?
2. How can L'Hôpital's Rule be applied to limits involving indeterminate forms?
3. What are the key trigonometric identities used in solving limits involving tangent functions?
4. Can we approach this problem using series expansions for the tangent function around $x = a$?
5. How does the behavior of $\tan(x)$ differ as $x$ approaches different multiples of $\frac{\pi}{2}$?

Tip: Always check if there are removable discontinuities (like common factors in the numerator and denominator) when simplifying limits.