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 ↓


Limits at Infinity


The concept of limits at infinity gives us important information about the behavior of a function as the input becomes very large, either positively or negatively. This is an important part of calculus, especially when analyzing functions and their asymptotic behavior.

What are the limitations?

Before learning about limits at infinity, it is important to understand what limits basically are. In calculus, the limit is the value that a function (or sequence) approaches as it approaches a value of the input (or index). It is represented as:

lim x→a f(x) = L

This expression states that as x approaches a, the function f(x) approaches the value L

Defining limits at infinity

Limits at infinity are specifically focused on describing the behavior of the function as the input becomes very large in a positive or negative sense. We define it as:

lim x→∞ f(x) = L

This means that as x increases without bound, the function f(x) approaches a specific value L Similarly, the notation:

lim x→−∞ f(x) = L

x describes the behavior of decreasing without bound.

Understanding through examples

Example 1: A rational function

Let us consider a simple rational function:

f(x) = 1/x

We are interested in finding the following:

lim x→∞ 1/x

As x gets larger, the fraction 1/x gets smaller and approaches 0 So:

lim x→∞ 1/x = 0
f(x) = 1/x

Example 2: Polynomial function

Polynomials also provide interesting information when we explore their limits at infinity. Consider the polynomial:

f(x) = 2x^3 - 3x^2 + x - 5

As x approaches infinity, the term 2x^3 dominates the behavior of the function because it grows faster than the other terms. Thus:

lim x→∞ 2x^3 - 3x^2 + x - 5 = ∞

Example 3: Exponential function

Now let's consider an exponential function:

f(x) = e^(-x)

Note for the limit as x approaches infinity:

lim x→∞ e^(-x) = 0

As x increases, e^(-x) decreases toward zero.

Finding limits at infinity: the analytical approach

Finding limits at infinity can sometimes be done by following a systematic procedure. Here we'll explore some of the main steps:

  • Determine the dominant term: Identify the fastest growing term in the function as x approaches infinity or negative infinity. In polynomials, this is the term with the highest power.
  • Simplify the function: Divide all terms by the leading term if necessary to simplify the limit calculation.

Let us apply this approach in the following examples:

Example 4: Complex rational function

Consider this function:

f(x) = (2x^2 + 3x + 1)/(5x^2 - 4x + 7)

we are interested in:

lim x→∞ (2x^2 + 3x + 1)/(5x^2 - 4x + 7)

The highest order term in both the numerator and denominator is x^2. Divide each term by x^2:

f(x) = (2 + 3/x + 1/x^2)/(5 - 4/x + 7/x^2)

As x → ∞, 3/x, 1/x^2, 4/x, and 7/x^2 all approach zero. So:

lim x→∞ f(x) = 2/5

Example 5: Limits at negative infinity

Consider:

f(x) = (7x^3 - 4x + 1)/(3x^3 + x - 2)

To find the limit at negative infinity, you'll follow similar steps. Divide all terms by x^3 and simplify:

f(x) = (7 - 4/x^2 + 1/x^3)/(3 + 1/x^2 - 2/x^3)

As x → −∞, 4/x^2, 1/x^3, 1/x^2, and 2/x^3 become zero:

lim x→−∞ f(x) = 7/3

The concept of horizontal asymptote

Limits at infinity often lead to identifying horizontal asymptotes. A horizontal asymptote is a horizontal line that a graph approaches as x goes to infinity or negative infinity. In mathematical terms, if:

lim x→∞ f(x) = L or lim x→−∞ f(x) = L

The line y = L is the horizontal asymptote of the graph of f(x).

Example 6: Rational function with horizontal asymptote

Celebration:

f(x) = (3x^3 + 5)/(2x^3 - x)

The highest order terms in the numerator are 3x^3 and in the denominator are 2x^3. Dividing both the numerator and the denominator by x^3, we get:

f(x) = (3 + 5/x^3)/(2 - 1/x^2)

Taking the limit as x approaches infinity or negative infinity:

lim x→∞ f(x) = 3/2
lim x→−∞ f(x) = 3/2

Here, both limits approach 3/2, so the function has a horizontal asymptote at y = 3/2.

Summary

Exploring limits at infinity helps us understand the long-term behavior of various mathematical functions. Through examples such as rational and polynomial functions, we have seen how these limits can be determined analytically, helping to highlight important features such as horizontal asymptotes.

Recognizing patterns, identifying key terms, and knowing when terms can be ignored are all useful skills when working with limits at infinity. By cementing these concepts, students can gain a deeper understanding of continuity and the behavior of functions in calculus.

This subject not only deals with basic limits and calculus but also extends to further studies in mathematical analysis, engineering, and applied sciences, where understanding the behavior of functions under extreme conditions plays a vital role.

Further practice with different types of tasks will reinforce these ideas, and prepare students for more advanced calculus topics and applications.


Grade 11 → 4.1.3


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One-Sided Limits
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Continuity of a Function

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Limits at Infinity

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