Introduction
Precalculus
Getting Started
Calculus 1
1. Limits
3. Derivatives
5. Applications of Derivatives
6. 1. Increasing and Decreasing Function
  2. Minimum and Maximum Values
7. Integrals
8. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
Calculus 2
1. Integration Techniques
2. 1. Integration by Parts
  2. Integrals Involving Trig Functions
  3. Trig Substitutions
  4. Partial Fractions
  5. Integrals Involving Roots
  6. Integrals Involving Quadratics
3. Parametric Equations and Polar Coordinates
4. 1. Differentiation Formulas
  2. Product and Quotient Rule
  3. Chain Rule
  4. Implicit Differentiation
  5. Higher Order Derivatives
  6. Logarithmic Differentiation
5. Series and Sequences
6. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
7. Vectors
8. 3-Dimensional Space
Calculus 3
1. LimitThree Dimensional Coordinate Systems
2. 1. Equations of Lines
  2. Equations of Planes
  3. Quadratic Surfaces
  4. Functions of Multiple Variables
3. Partial Derivatives
4. 1. Higher Order Partial Derivatives
  2. Differentials
  3. Chain Rule
  4. DirectioChain Rulenal Derivatives
  5. Gradient
5. Multiple Integrals
6. 1. Iterated Integrals in Cylindrical Coordinates
  2. Double Integrals
  3. Double Integrals in Polar Coordinates
  4. Triple Integrals in Spherical Coordinates
  5. Change of Variables
  6. Surface Area
7. Line Integrals
8. 1. Line Integrals With Respect to Arc Length
  2. Double Integrals in Polar Coordinates
  3. Line Integrals With Respect to x and y
  4. Line Integrals of Vector Fields
  5. Fundamental Theorem of Line Integrals
  6. Conservative Vector Fields
  7. Potential Functions
  8. Green's Theorem
9. Surface Integrals
10. 1. Parametric Surfaces
  2. Surface Integrals
  3. Surface Integrals of Vector Fields
  4. Stokes' Theorem
  5. Divergence Theorem
Cheatsheet

One-Sided Limits

What is One-Sided Limits?

One-sided limit is a type of limit that describes how a function behaves as the input approaches a given value from one side only. This is different from a standard limit, which describes how a function behaves as the input approaches a value from both sides.

One-sided limits are often used to describe how a function behaves at points where it is not defined or is not continuous. For example, if a function has a "jump" or a "break" at a certain point, the one-sided limits of the function at that point will be different depending on which side of the point the input is approaching from.

Mathematically, a one-sided limit of a function $f(x)$ at a point a is defined as follows:

$$ \lim_{x \to a^-} f(x) = L $$

This equation states that the one-sided limit of the function $f(x)$ at the point a from the left (i.e., as x approaches a from values less than a) is equal to the value L. Similarly, the one-sided limit of the function at the point a from the right can be written as follows:

$$ \lim_{x \to a^+} f(x) = L $$

This equation states that the one-sided limit of the function $f(x)$ at the point a from the right (i.e., as $x$ approaches a from values greater than a) is equal to the value L.

If the one-sided limits from both sides are equal, then the function is considered continuous at the point a.
If the one-sided limits from both sides are not equal, then the function is considered discontinuous means the Limit doesn't exists.

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