Describe the concept of recursion in Python. Provide examples of recursive functions and discuss their advantages and limitations.

Recursion in Python is a programming technique where a function calls itself repeatedly to solve a problem by breaking it down into smaller, similar subproblems. In recursive functions, the solution to a problem depends on solutions to smaller instances of the same problem.

The key components of a recursive function are the base case and the recursive case. The base case represents the simplest form of the problem that can be solved directly without further recursion. The recursive case defines how the problem is broken down into smaller subproblems and how the recursive function is called on those subproblems.

Let's consider an example of a recursive function that calculates the factorial of a number:

```
python`def factorial(n):
if n == 0:
return 1
else:
return n * factorial(n - 1)`
```
In this example, the base case is when `n` is 0, in which case the function returns 1. The recursive case calculates the factorial of `n` by multiplying `n` with the factorial of `n-1`. The function keeps calling itself with a smaller value until it reaches the base case.

Advantages of Recursive Functions:

1. Clarity and Readability: Recursive functions can often provide a more intuitive and concise solution to a problem, especially when the problem can be naturally divided into smaller subproblems.
2. Solving Complex Problems: Recursive functions are particularly useful for solving problems that can be divided into smaller, identical subproblems. They enable a divide-and-conquer approach to problem-solving, making it easier to reason about and implement solutions.
3. Recurring Patterns: Certain problems inherently exhibit recurring patterns, and recursive functions allow us to leverage these patterns in a natural and elegant way.
4. Mathematical Algorithms: Many mathematical algorithms, such as Fibonacci sequences, tree traversals, and sorting algorithms like quicksort or mergesort, are naturally expressed using recursion. Recursive implementations can often closely resemble the mathematical description of the problem, making them easier to understand and analyze.

Limitations of Recursive Functions:

1. Performance Overhead: Recursive functions can be less efficient compared to iterative solutions due to the overhead of function calls and managing the call stack. Each recursive call adds a new frame to the call stack, potentially leading to stack overflow errors if the recursion depth is too large.
2. Space Complexity: Recursive functions can consume a significant amount of memory if the recursion depth is large. Each function call adds a new stack frame, and if the number of recursive calls is high, it can lead to excessive memory usage.
3. Lack of Tail Recursion Optimization: Python does not optimize tail recursion, which means that recursive functions with tail recursion may consume more stack space compared to equivalent iterative solutions.
4. Code Readability: While recursion can provide elegant solutions, it can also be more challenging to understand and debug compared to iterative approaches. Recursive functions may require careful reasoning about the base case, recursive case, and termination conditions.

When using recursive functions, it's essential to ensure that the base case is well-defined and that there is a termination condition to prevent infinite recursion. Additionally, for performance-sensitive applications, it may be necessary to consider iterative or optimized solutions that avoid excessive stack usage.

In summary, recursion is a powerful technique in Python for solving problems by breaking them down into smaller, similar subproblems. Recursive functions can provide clear and concise solutions, especially for problems with recurring patterns or mathematical algorithms. However, they come with performance and memory overhead and require careful handling of base cases and termination conditions.