Understanding Limits

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 $x$ approaches $a$ is $L$."

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 $x^2$ will get closer to $2^2 = 4$.

Think of $x$ as a car. If we're getting closer to 2, what number is $x^2$ approaching? Like the car nearing the red light, $x^2$ gets closer to 4 as $x$ approaches 2.

We can calculate this practically using a small value $h$: $f(x + h)$ Here, $h$ is a very small number.

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

# Numeric limit calculation
def limit(f, x, h=1e-5):
    return f(x + h)

# Sample function: f(x) = x^2
f = lambda x: x**2

# Compute limit of f at x=2
print("Limit of f(x) as x approaches 2:", limit(f, 2))  # Limit of f(x) as x approaches 2: 4.00004

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 $x$ approaches 2 and print the result.