Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus → Limits and Continuity ↓


One-Sided Limits


Understanding the concept of one-sided limits is important in the study of calculus. One-sided limits allow us to explore the behavior of functions as they approach a specific point from one side—either to the left or to the right. This concept can be especially useful for discontinuous functions or at points where functions change very rapidly.

Introduction to one-sided limits

In mathematics, a one-sided limit refers to the limiting behavior of a function as the input or variable approaches a point from only one side. Formally, there are two types of one-sided limits:

  • The left-hand limit, denoted as lim x→c⁻ f(x), refers to the value that the function f(x) approaches as x approaches c from the left.
  • The right-hand limit, denoted lim x→c⁺ f(x), refers to the value that the function f(x) approaches as x approaches c from the right.

In simple terms, one-sided limits find out how close a function approaches a particular value when approached from a particular direction, helping us understand functions that exhibit different behaviour from the right and left of a point.

Visual example

To understand one-sided limits more clearly, consider the following graphical representation:

Suppose we have a function f(x) where:

C f(x)

In the above graph, you can see a function f(x) graphed in blue that has a discontinuity at the point x = c.

  • The left limit can be seen as x approaches c from the left, that is, lim x→c⁻ f(x) = L₁.
  • The right-hand limit can be understood as the limit as x increases from the right towards c, that is, lim x→c⁺ f(x) = L₂.

If L₁ equals L₂, then a two-sided limit exists and, lim x→c f(x) = L.

Textual examples

Example 1: Piecewise function

Suppose you have a piece-wise function defined as:

f(x) = { 2x + 3, if x < 4 5, if x = 4 3x - 1, if x > 4 }

For x = 4, let's analyze the one-sided limits:

  • Left limit: lim x→4⁻ (2x + 3) = 2(4) + 3 = 11
  • Right-hand limit: lim x→4⁺ (3x - 1) = 3(4) - 1 = 11

Although the value of the function is f(4) = 5, the limits on both sides coincide, lim x→4 f(x) = 11, which is different from the value 4 of the function due to the piecewise nature of the definition.

Example 2: Rational function

Consider the rational function:

f(x) = (x^2 - 9) / (x - 3)

Note that the denominator becomes 0 when x = 3. We analyze one-sided limits:

  • Left-hand limit: (x^2 - 9) / (x - 3) simplifies to x + 3 for x ≠ 3. Continuing from the left, lim x→3⁻ (x + 3) = 6.
  • Right-hand limit: Similarly, lim x→3⁺ (x + 3) = 6 as we come from the right.

Here, both limits show that although the function is undefined at x = 3, both one-sided limits point to 6, which confirms that the limit of f(x) as x approaches 3 is also 6.

One-sided limits and continuity

One-sided limits play an important role in understanding function continuity. For a function to be continuous at a point c, the following must be true:

  • The function f(x) must be defined at x = c.
  • The limit of f(x) as x approaches c must exist.
  • The value of the limit must be equal to the value of the function at that point, which means lim x→c f(x) = f(c).

If the one-sided limits at some point c are divergent, then the limit at c does not exist, which indicates discontinuity.

Why one-sided limits are useful

In calculus, one-sided limits solve problems by giving us tools to investigate complex function behaviors, especially at discontinuities and jumps. We can evaluate real-world phenomena such as signal switching in electronics, material stress points in physics, and optimizing systems in engineering with precision.

Conclusion

One-sided limits are a foundational concept in calculus that enables us to analyze and understand the approaches of functions in specific directions. They are essential in studying continuity and understanding functions that may seem arbitrary or unpredictable at first glance. Mastery of one-sided limits equips students with the analytical skills needed for more advanced mathematics, setting an important cornerstone in their educational journey.


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One-Sided Limits

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