The given expression is:

$\pi + \frac{3(\infty)}{\pi(\infty) - 3}$

Step-by-step analysis:

1. The term $\infty$ (infinity) is involved, which often leads to special considerations in mathematics, especially when operations like division or subtraction with infinity are involved.
2. Let's break down the fraction:

$\frac{3(\infty)}{\pi(\infty) - 3}$

In this case, we have infinity in both the numerator and the denominator.

• The numerator is $3(\infty) = \infty$.
• The denominator $\pi(\infty) - 3$ simplifies to $\infty - 3 = \infty$, since subtracting any finite number from infinity still results in infinity.

So, the fraction becomes:

$\frac{\infty}{\infty}$

Indeterminate Form:

The expression $\frac{\infty}{\infty}$ is indeterminate, meaning it doesn't have a well-defined value without further context or limits. Without additional information, this expression is undefined.

Finally, adding $\pi$ to an undefined or indeterminate value will still result in an undefined or indeterminate result.

Conclusion:

The expression as it stands is undefined due to the $\frac{\infty}{\infty}$ indeterminate form.

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Would you like more details on handling limits and indeterminate forms, or do you have any further questions?

Related Questions:

1. What are indeterminate forms in calculus, and how are they resolved?
2. How do you compute limits involving infinity in fractions?
3. What happens when you add a finite number to infinity?
4. Can $\frac{\infty}{\infty}$ ever be evaluated in certain contexts?
5. How does L'Hopital's Rule help in dealing with indeterminate forms?

Tip: In calculus, indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ are often resolved using limit techniques like L'Hopital's Rule.