Function caching in python python tutorial day 92

## function caching in python

function caching is a technique used to store the results of expensive function calls and reuse them when the same inputs occur again. this can significantly speed up the performance of programs, especially when dealing with functions that involve heavy computations or resource-intensive operations.

### why use caching?

- **performance improvement**: reduces the number of times a function needs to be executed.
- **resource efficiency**: saves cpu time and reduces load on resources.
- **simplifies code**: can sometimes help in reducing complexity in code by avoiding repetitive calculations.

### python's built-in caching

python provides a built-in decorator called `lru_cache` (least recently used cache) in the `functools` module that makes caching easy. it is particularly useful for functions that return the same output for the same input.

### how to use `lru_cache`

1. **import the `lru_cache` decorator** from the `functools` module.
2. **apply the decorator to a function** you want to cache.
3. **call the function normally** and enjoy the benefits of caching.

### example

let's create an example where we compute fibonacci numbers, which is a classic case that benefits from caching due to its recursive nature.

### explanation of the example

1. **importing `lru_cache`**: we import the necessary decorator from the `functools` module.
2. **defining the fibonacci function**: the `fibonacci` function is defined to compute fibonacci numbers recursively.
4. **testing the function**: when we call `fibonacci(n)` with larger values of `n`, the function will run faster after the first call because it will retrieve results from the cache instead of recalculating them.

### cache management

you can manage the cache with the following methods:

- `cache_c ...

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