Welcome to our lesson on understanding limits, a fundamental concept in calculus with important applications in machine learning. Limits help us understand how functions behave as we approach certain points, which is crucial for defining derivatives and integrals. By the end, you'll grasp what limits are, learn how to compute them numerically, and see how to implement this in Python.

A limit in mathematics describes the value a function approaches as the input nears some value. Imagine driving toward a red light. As you get closer, you're approaching a specific point where you'll stop. That's similar to limits: as the input nears a particular value, the output of the function approaches a specific value.

$\lim_{{x \to a}} f(x) = L$ This reads as "the limit of $f(x)$ as approaches is ."

Consider the function $f(x) = x^2$. To find the limit as $x$ approaches 2, think about what happens to $f(x)$ when $x$ gets very close to 2. The values of will get closer to .

Let's see how to calculate limits numerically using Python.

1. Define Limit Function: The limit function calculates the function value for a small increment $h$. Here, f is the function, x is the point, and h is a small number.
2. Define Sample Function: We use a lambda function $f(x) = x^2$ to keep it simple.
3. Compute and Print: Compute the limit as approaches 2 and print the result.

The choice of $h$ is crucial. It needs to be small but not too small to avoid numerical errors due to computer precision. Typically, $h$ is set around $10^{-5}$ or smaller for good accuracy.

Consider this: if you’re too close to the wall (like $h$ being too small), you can't see much of the wall's shape. Step a tiny bit back (a reasonable $h$), and the shape becomes clearer without much distortion.

Let's examine the function $\frac{\sin(x)}{x}$, which is undefined at $x = 0$. This is because division by zero is not allowed in mathematics.

Here is the graph of $$:

The limit is: $\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1$

Knowing this, we can redefine our function:

We've covered what limits are, why they are important, and how to calculate them both conceptually and numerically. We broke down a Python code snippet demonstrating this process for the function $f(x) = x^2$. Remember, limits help understand how functions behave as inputs get close to a particular point, which is key for more advanced topics like derivatives.

Now it's time to put your new knowledge into practice! You'll be working on hands-on exercises where you'll code your own limit functions and evaluate limits for different functions and points in the CodeSignal IDE. Happy coding!