Summary  
This chapter explains the concept of limits, covering one- and two-sided limits, methods for evaluating them (direct substitution, factoring and canceling, rationalization, special limits), and the common failure cases (jump discontinuities, infinite limits, oscillations).

General domain of usage  
Calculus

**A limit** is a fundamental concept in calculus that describes the value a function approaches as its input nears a specific point. Limits form the basis for defining derivatives and integrals, making them essential in mathematical analysis and machine learning optimization.


Definition

## Formal Definition & Notation
A limit represents the value that a function approaches as the input gets arbitrarily close to a point.

$$
\lim_{x \rarr a}f(x) = L
$$

This means that as $$x$$ gets arbitrarily close to $$a$$ where    approaches $$L$$.

The function does **not** need to be defined at $$x=a$$ for the limit to exist.

Note

## One-Sided & Two-Sided Limits
A limit can be approached from either side:
- **Left-hand limit**:
  approaching $$a$$ from values smaller than $$a$$:
  $$
  \lim_{x \rarr a^-}f(x)
  $$
- **Right-hand limit**: approaching $$a$$ from values larger than $$a$$:
  $$
  \lim_{x \rarr a^+}f(x)
  $$
- The limit exists only if **both one-sided limits are equal**:
  $$
  \lim_{x \rarr a^-}f(x) = \lim_{x \rarr a^+}f(x)
  $$
   


## When Limits Fail to Exist
A limit does **not** exist in the following cases:
- **Jump discontinuity**:
  $$
  \lim_{x \rarr a^-}f(x) \neq \lim_{x \rarr a^+}f(x)
  $$
  - **Example**: a step function where the left and right limits are different.
- **Infinite limit**:
  $$
  \lim_{x \rarr 0}\frac{1}{x^2}=\infty
  $$
  - The function grows unbounded.
- **Oscillation**:
  $$
  \lim_{x \rarr 0}\sin\left(\frac{1}{x}\right)
  $$
  - The function fluctuates infinitely without settling to a single value.



## Special Case – Limits at Infinity
When $$x$$ approaches infinity, we analyze the **end behavior** of functions:
- **Rational functions**:
  $$
  \lim_{x \rarr \infty}\frac{1}{x}=0
  $$
- **Polynomial growth**:
  $$
  \lim_{x \rarr \infty}\frac{x^2}{x}=\infty
  $$
- **Dominant term rule**:
  $$
  \lim_{x \to \infty} \frac{a x^m}{b x^n} = \begin{cases} 0,\ \text{if } m < n,\\ \frac{a}{b},\ \text{if } m = n, \\ \pm \infty,\ \text{if } m > n. \end{cases}
  $$



Which statement correctly describes when a limit exists?

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Introduction to Limits

Formal Definition & Notation

One-Sided & Two-Sided Limits

When Limits Fail to Exist

Special Case – Limits at Infinity