Limit of a Function

Yaser Rahmati | یاسر رحمتی

Definition

Let $f(x)$ be a function defined for all $x$ near $c$ (except possibly at $c$ itself). We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$, written as:

$$\lim_{{x \to c}} f(x) = L$$

if for every positive number $ε$, there exists a positive number $δ$ such that if $0<∣x−c∣<δ$, then $∣f(x)−L∣<ε$.

Understanding the Concept

In simpler terms, it means that as $x$ gets arbitrarily close to $c$, $f(x)$ gets arbitrarily close to $L$.

Example 1

$$\lim_{x\to 2} (x^2 + 3x - 2)$$

To evaluate this limit, you can simply plug in (x = 2) into the function:

$$2^2 + 3 \cdot 2 - 2 = 4 + 6 - 2 = (8). So, \lim_{x\to 2} (x^2 + 3x - 2) = 8$$

import sympy as sp

# Define the variable and the function
x = sp.symbols('x')
f = x**2 +3*x - 2

# Calculate the limit as x approaches 3
limit_value = sp.limit(f, x, 3)
print(limit_value)

Example 2

$$\lim_{x\to 0} \frac{\sin(x)}{x}$$

This is a classic example in calculus. As (x) approaches 0, the function $\frac{\sin(x)}{x}$approaches 1.

import sympy as sp

# Define the variable and the function
x = sp.symbols('x')
f = sp.sin(x) / x

# Calculate the limit as x approaches 0
limit_value = sp.limit(f, x, 0)
print(limit_value)

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