The given mathematical expression is:

$\lim_{a \to -\infty} \arctan x \bigg|_{a}^{0}.$

This represents the evaluation of the definite limit of the difference of the arctangent function from $x = a$ to $x = 0$, as $a \to -\infty$.

1. The function $\arctan x$ evaluates to:

   • At $x = 0$: $\arctan(0) = 0$.
   • At $x = a$: $\arctan(a)$.

2. Substituting these into the limit, the expression simplifies to: $\lim_{a \to -\infty} [\arctan(0) - \arctan(a)].$

3. As $a \to -\infty$, the behavior of $\arctan(a)$ is:

   • $\arctan(a) \to -\frac{\pi}{2}$, since the arctangent function approaches $-\frac{\pi}{2}$ for very large negative values of $a$.

4. Therefore: $\lim_{a \to -\infty} [0 - \arctan(a)] = \lim_{a \to -\infty} -\arctan(a).$

5. Simplify further: $\lim_{a \to -\infty} -(-\frac{\pi}{2}) = \frac{\pi}{2}.$

Final Answer:

$\lim_{a \to -\infty} \arctan x \bigg|_{a}^{0} = \frac{\pi}{2}.$

Would you like a detailed explanation of how $\arctan$ behaves at extreme values or any clarification on these steps?

Here are 5 relative questions for deeper exploration:

1. What are the general properties of the arctangent function?
2. How does the concept of a definite integral relate to limits in this context?
3. Why does $\arctan(x)$ have horizontal asymptotes at $\pm \frac{\pi}{2}$?
4. Can you compute similar limits for other inverse trigonometric functions like $\arcsin(x)$?
5. What happens if the limits are taken symmetrically around zero, such as $\lim_{a \to \infty} \arctan(x)|_{-a}^{a}$?

Tip:

Always graph the function when dealing with limits involving asymptotes or extreme values to get an intuitive understanding!