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Graphing Calculator by Mathlab: User Manual
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  • Home
    • Introduction
    • PRO Features vs. FREE Version
    • Frequently Asked Questions, FAQs >
      • 1. How to Change the Number Format?
      • 2. How to Set Up the Separators Between Thousands?
      • 3. How to Set Precision?
      • 4. How to Send Feedback with Comments?
      • 5. How to import/export the library?
      • 6. How to Print Results?
      • 7. How to Make the Calculator Show the Results?
      • 8. How to Transport Calculation Results to other Programs?
      • 9. How to Transport Table to Other Platforms?
      • 10. How to Turn Off (or on) Vibration?
      • 11. How to Change the Language?
  • 1. Basics
    • 1.1. Navigation
    • 1.2. UI Elements
    • 1.3. Keyboard
    • 1.4. Input, Enter, Delete, Clear and UNDO Buttons
    • 1.5. Workspace Area
    • 1.6. Editing the Expression/Equation
    • 1.7. Using the Last Answer
    • 1.8. Writing Comments
    • 1.9. Clear, Copy & Paste Commands
    • 1.10. Rearranging Rows
  • 2. Settings
    • 2.1. General
    • 2.2. Calculator
    • 2.3. Graph
  • 3. Library
    • 3.1. Constants
    • 3.2. Functions
    • 3.3. How to Save Calculation Result/Graph to Library?
  • 4. Graph Mode
    • 4.1. 2D Graphing
    • 4.2. 3D Graphing
    • 4.3. Enlarging the Graph Area
    • 4.4. Changing to White Background
    • 4.5. Hide Keyboard
    • 4.6. Degree and Radian Scales
    • 4.7. Fixed Scale
    • 4.8. R-axis Scale
    • 4.9. Logarithmic Scale
    • 4.10. Tracing Values and Slopes
    • 4.11. Special Points: Roots and Criticals
    • 4.12. Intersections of Graphs
    • 4.13. Set Domain
    • 4.14. Show All - Roots, Critical Points and Intersections
    • 4.15. Fullscreen
  • 5. Table Mode
    • 5.1. Sharing of Functions
    • 5.2. 2D Table
    • 5.3. 3D Table
    • 5.4. Edit Functions
    • 5.5. Scroll Results
    • 5.6. Results Precision
    • 5.7. Zoom Controls
    • 5.8. Save and Load Table
    • 5.9. Table of Trigonometric Functions
  • 6. Numbers and Number Sense
    • 6.1. Decimals
    • 6.2. Fractions >
      • 6.2.1. Mixed Fractions
      • 6.2.2. Complex Fractions
      • 6.2.3. Converting Decimals to Fractions
      • 6.2.4. Converting Fractions to Decimals
    • 6.3. Percents
    • 6.4. Scientific Notation
    • 6.5. Engineering Notation
    • 6.6. Rounding Numbers
    • 6.7. Integer and Fractional Parts >
      • 6.7.1. Integer Part of a Number >
        • 6.7.1.1. Ceiling
        • 6.7.1.2. Floor
        • 6.7.1.3. Half Down
        • 6.7.1.4. Half to Even
        • 6.7.1.5. Half to Infinity
        • 6.7.1.6. Half to Odd
        • 6.7.1.7. Half to Zero
        • 6.7.1.8. Half Up
        • 6.7.1.9. Truncate
      • 6.7.2. Greatest Integer is the Floor Function
      • 6.7.3. Least Integer is the Ceiling Function
      • 6.7.4. Fractional Part of a Number
    • 6.8. Order of Operations
    • 6.9. Least Common Multiple
    • 6.10. Greatest Common Divisor
    • 6.11. Modulo
    • 6.12. Binary, Octal, Decimal, Hexadecimal Numbers
    • 6.13. Complex Numbers
    • 6.14. The Polar Form of Complex Numbers
    • 6.15. Polar to Rectangular Coordinates
  • 7. Introductory Algebra
    • 7.1. Arithmetic Operations
    • 7.2. Exponents
    • 7.3. Absolute Values
    • 7.4. Variables
    • 7.5. Evaluating Expressions
    • 7.6. Polynomials
    • 7.7. Roots
    • 7.8. Logarithms
  • 8. Equations in One Variable
    • 8.1. Linear Equation
    • 8.2. Absolute Value Equation
    • 8.3. Quadratic Equation
    • 8.4. Cubic Equation
    • 8.5. Polynomial Equation
    • 8.6. Rational Equation
    • 8.7. Radical Equation
    • 8.8. Exponential Equation
    • 8.9. Logarithmic Equation
  • 9. Inequalities in One Variable
    • 9.1. Inequality Symbols
    • 9.2. Linear Inequalities
    • 9.3. Absolute Value Inequalities
    • 9.4. Quadratic Inequality
    • 9.5. Polynomial Inequalities
    • 9.6. Rational Inequalities
    • 9.7. Compound Inequalities
    • 9.8. Inequalities with Constants
  • 10. Equations and Inequalities in Two Variables
    • 10.1. Linear Equations
    • 10.2. Systems of Linear Equations
    • 10.3. Graphing Inequalities
    • 10.4. Multiple Graphing of Inequalities
    • 10.5. Graphing Systems of Inequalities
    • 10.6. Solving Implicit Equations
  • 11. Algebraic Functions and Graphs
    • 11.1. Plotting Points
    • 11.2. How to Graph Functions?
    • 11.3. Setting the Applied Domain
    • 11.4. Linear Function
    • 11.5. Absolute Value Function
    • 11.6. Quadratic Function
    • 11.7. Polynomial Functions
    • 11.8. Rational Functions
    • 11.9. Radical Functions
    • 11.10. Logarithmic Functions
    • 11.11. Exponential Functions
    • 11.12. Sign Function
    • 11.13. Multiple Graphing
    • 11.14. Piecewise Functions
  • 12. Matrices and Vectors
    • 12.1. Matrix Operations
    • 12.2. Editing Matrix Entries
    • 12.3. Matrix Variables
    • 12.4. Matrix and Vector Forms
    • 12.5. Variable Matrix to System of Linear Equations
    • 12.6. Solving Systems of Linear Equations Using Matrix Equations
  • 13. Trigonometric Functions and Their Inverses
    • 13.1. Degrees and Radians >
      • 13.1.1. Degrees, Minutes and Seconds
      • 13.1.2. Bradis Table
    • 13.2. Trigonometric Function Keys
    • 13.3. Trigonometric Values of Special Angles >
      • 13.3.1. The 45- 45 - 90 Triangle
      • 13.3.2. The 30-60-90 Triangle
      • 13.3.3. Quadrantal Angles
      • 13.3.4. Coterminal Angles
    • 13.4. Trigonometric Values of 15 Degrees and Its Multiples
    • 13.5. Hyperbolic Function Keys
    • 13.6. Graphing Trigonometric Functions
    • 13.7. Graphing Hyperbolic Functions
    • 13.8. Graphing Inverse Functions
  • 14. Analytic Geometry
    • 14.1. Conic Sections
    • 14.2. Parametric Equations
    • 14.3. Polar Graphs >
      • 14.3.1. Limacon
      • 14.3.2. Cardioid
      • 14.3.3. Lemniscate
      • 14.3.4. Rose
      • 14.3.5. Other Polar Graphs
    • 14.4. 3D Graphing
  • 15. Limits
    • 15.1. Right - hand Limit
    • 15.2. Left - hand Limit
    • 15.3. Limit of a Function
    • 15.4. Limit of a Polynomial Function
    • 15.5. Limit of a Rational Function
    • 15.6. Limit of a Radical Function
    • 15.7. Limit of an Absolute Value Function
    • 15.8. Limit of a Trigonometric Function
    • 15.9. Limit of an Exponential and Logarithmic Function
    • 15.10. Limit of a Piece-wise Function
    • 15.11. Limits at Infinity
    • 15.12. Indeterminate Forms
    • 15.13. Limit of a Hyperbolic Function
  • 16. Derivatives
    • 16.1. First Derivative Key
    • 16.2. Second Derivative Key
    • 16.3. Third and Higher Derivative Keys
    • 16.4. Rules of Differentiation
    • 16.5. Derivatives of Polynomial Functions
    • 16.6. Derivatives of Rational Functions
    • 16.7. Dervatives of Trigonometric, Logarithmic and and Exponential Functions
    • 16.8. More on Derivatives
  • 17. Partial Derivatives
    • 17.1. Increments
    • 17.2. Dervative of a Function df (or dy))
    • 17.3. Derivative of a Function df (f not in terms of x)
    • 17.4. Partial Derivatives
    • 17.5. Higher Order Partial Derivatives
    • 17.6. Total Derivates
  • 18. Definite Integral
    • 18.1. Definite Integral of Algebraic Functions
    • 18.2. Definite Integral of Trigonometric Functions
  • 19. Basic Statistics
    • 19.1. Summation Notation
    • 19.2. Product Notation
    • 19.3. Minimum and Maximum
    • 19.4. Factorial, nCr and nPr
    • 19.5. Measures of Central Tendency >
      • 19.5.1. Mean from Ungrouped Data Set
      • 19.5.2. Mean from Frequency Distribution Table
      • 19.5.3. Median from Ungrouped Data Set
      • 19.5.4. Mode
    • 19.6. Measures of Variability >
      • 19.6.1. Range
      • 19.6.2. Interquartile Range and Quartile Deviation
      • 19.6.3. Mean Absolute Deviation
      • 19.6.4. Variance and Standard Deviation
      • 19.6.5. Coefficient of Variation
    • 19.7. Measures of Position
    • 19.8. Bivariate Data Analysis >
      • 19.8.1 Covariance
      • 19.8.2. Correlation Coefficient
      • 19.8.3. Scatterplot and Regression Line
  • 20. Special Functions
    • 20.1. Gamma Function
    • 20.2. Logarithmic Gamma Function
    • 20.3. Digamma Function
  • 21. List of ALL Functions
    • 21.1. Arithmetics
    • 21.2. Algebra
    • 21.3. Trigonometry
    • 21.4. Statistics
    • 21.5. Calculus

15.11. Limit at Infinity

Polynomial Functions
Finding the limit of a polynomial function at infinity is the same as finding the limit as x approaches +∞. Similarly, the limit at negative infinity is the same as finding the limit as x approaches -∞. If f(x) increases as x increases then the limit of f(x) as x approaches +∞ is +∞. If f(x) decreases as x approaches -∞, then the limit of f(x) as x approaches -∞ is  -∞.

You can determine the limit of a function as it approaches -∞ or +∞ based on its degree (the highest exponent).

1) When the degree is odd, such as f(x) = x or f(x) = x^3

Case 1:  a > 0   
    The limit at +∞ is +∞.
    The limit at -∞ is -∞.

Case 2:  a < 0   
     The limit at +∞ is -∞. 
     The limit at -∞ is +∞.
 
2) When the degree is even, such as f(x) = x^2 or f(x) = x^4

Case 1:  a > 0   
    The limit at +∞ is +∞.
    The limit at -∞ is +∞.     
Case 2:  a < 0   
    The limit at +∞ is -∞.
    The limit at -∞ is -∞.    

Illustrative Example
Find the limit of f(x) = x^3 - 4x^2 + 6 as x approaches -∞.
This is a 3rd degree polynomial with a = 1 > 0. So the limit as x approaches -∞ is -∞.

Calculator solution
Step 1: Tap and hold the exponent key. Select "lim." You can also tap the exponent key four times.
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Step 2: Enter infinity by holding the degree key and selecting ∞. You can also tap the degree key four times.
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Step 3: Enter the function in parentheses.
Type in: lim [ x = -∞ ] ( 9x^3 - 4x^2 + 6 )
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More Examples
Find the limit at infinity for each polynomial function below.
Calculator solution
​Type in: lim [ x = ∞ ] 3
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Calculator solution
​Type in: lim [ x = ∞ ] ( 3x^5 - 4x + 2x - 3 )
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Calculator solution
​Type in: lim [ x = - ∞ ] ( -3x^4 - 4x^3 + 2x - 3 )
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Calculator solution
​Type in: lim [ x = ∞ ] ( x^2 - x^4 )
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Rational Functions
To find the limit at infinity of a rational function, let ax^n be the first term of the numerator and bx^m be the first term of the denominator.

1) If the degree of the numerator is equal to the degree of the denominator, the limit at infinity is a/b. In the example below, the degrees are the same ( x^3 ), so the limit at infinity is 4/2 = 2.
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2) If the degree of the numerator is less than the degree of the denominator, then the limit at infinity is 0. In the example below, the degree of the numerator is x^2 which is less than the degree of the denominator x^3, so the limit is 0.
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3) If the degree of the numerator is greater than the denominator, the limit is either positive or negative infinity.
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More Examples
Calculate the limit at infinity for each rational function below.
Calculator solution
​Type in: lim [ x = ∞ ] ( ( x^4 - 10 ) / ( 4x^3 + x ) )
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Calculator solution
​Type in: lim [ z = -∞ ] ( ( 4z^2 + z ^ 6 ) / ( 1 + 5z^3 ) )
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Calculator solution
​Type in: lim [ x = -∞ ] ( ( 2x^2 ) / ( x^2 - 4 ) )
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Calculator solution
​Type in: lim [ x = ∞ ] ( ( 4x^3 ) / ( 5x^2 - 3 ) )
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Calculator solution
​Type in: lim [ x = ∞ ] ( ( 5x^2 - 7x + 9 ) / ( x^2 -2x - 3 ) )
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Calculator solution
​Type in: lim [ x = ∞ ] ( ( 3x^2 - 7x ) / ( x - 8 ) )
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Radical Functions
To find the limit at infinity of a radical function, you can substitute large values of x to see the behavior of the function f(x). The limit also depends on the degree of the radical and the radicand. If the radicand contains a fraction, the rules for rational expressions hold. 

Illustrative Examples
Calculate the limit at infinity for each radical function below.
As x increases, the value of sqrt(3x - 2) increases. So the limit is +∞.
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The domain of the radical function is the set of real numbers greater than or equal to 3/2, so the limit as x approaches -∞ does not exist.
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The function is positive for any x, so plugging in a large negative number into f(x) would result in a large positive number. The limit     is +∞.
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More Examples
Calculate each limit at infinity below.
Calculator solution
Type in: lim [ x = -∞ ] cbrt( ( x - 3 ) / ( 5 - x ) )
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Calculator solution
Type in: ​lim [ x = ∞ ] ( x + sqrt( x^2 + 2x ) )
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Calculator solution
Type in: ​lim [ x = ∞ ] ( ( sqrt x - 3 ) / ( x - 9 ) )
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Calculator solution
Type in: ​lim [ x = ∞ ] ( sqrt( x^6 + 3x^2 + 1 ) / ( 4x^3 + 3 ) )
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Calculator solution
Type in: ​lim [ x = ∞ ] ( sqrt( x^2 + 3 ) / ( 7x + 5 ) )
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Absolute Value Functions
The graph of an absolute value function may increase or decrease at all values of x depending on whether it opens upward or downward.
If a > 0, the graph opens upward. As x increases, f(x) increases. As x decreases, f(x) also increases. The limit at infinity is always +∞.

If a < 0, the graph opens downward. As x increases, f(x) decreases. As x decreases, f(x) also decreases. The limit at infinity is -∞.
Examples
Calculate each limit at infinity if it exists.
Calculator solution
Type in: lim [ x = -∞ ] abs ( 2x + 3 ) - 3
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Calculator solution
Type in: ​lim [ x = -∞ ] ( -abs ( 3x + 2 ) + 2 )
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Calculator solution
Type in: ​lim [ x = ∞ ] (2 ( abs( x^2 - 4 ) - abs( x ) - 4 ) )
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Calculator solution
Type in: ​lim [ x = -∞ ] ( 2 / abs( 2x - 1 ) )
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Calculator solution
Type in: ​lim [ x = -∞ ] ( ( x - 1 ) / abs( x - 2 ) )
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Exponential Functions
An exponential function with a base b > 1 has a graph that increases as x increases, so the limit at positive infinity is +∞. Note that an exponential function f(x) = b^x also has an asymptote. The asymptote is the limit at negative infinity.

When 0 < b < 1, the reverse is true. The limit at positive infinity is the y-value of the asymptote while the limit at negative infinity is +∞.

Illustrative Examples
Calculate each limit at infinity.
Since b = 3 > 1, the limit at infinity is +∞.
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Since b = 1/2 < 1, f(x) approaches the asymptote y = 3. The limit at positive infinity is 3.
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More Examples
Calculate each limit below if it exists.
Calculator solution
Type in: lim [ x = ∞ ] ( 2^( x - 2 ) )
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Calculator solution
​Type in: ​lim [ x = -∞ ] ( 2^( x - 2 ) )
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Calculator solution
​Type in: ​lim [ x = -∞ ] ( 2 ^( x - 2 ) + 3 )
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Calculator solution
​Type in: ​lim [ x = -∞ ] ( -e^( -2x ) + 3 )
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Logarithmic Functions
A logarithmic function is the inverse of an exponential function. Since the range of an exponential function is the set of real numbers greater than zero, the domain of a logarithmic function is the set of real numbers greater than zero. Therefore evaluating limits at negative infinity is not possible. A logarithmic function also has a vertical asymptote. The expression and base of the logarithmic function determine whether it approaches negative or positive infinity.

Illustrative Examples
Calculate each limit at infinity if it exists.
As x increases, f(x) increases, so the limit is +​∞.
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Because the domain must be positive, the limit at negative infinity is undefined.
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More Examples
Calculate each limit at infinity if it exists.
Calculator solution
Type in: lim [ x = ∞ ] log ( 2x )
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Calculator solution
​Type in: ​lim [ x = ∞ ] log ( 2x^2 - 3 )
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Calculator solution
​Type in: ​lim [ x = ∞ ] ( log ( 2x^3 - 3 ) + 2 ln x )
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Calculator solution
​Type in: ​lim [ x = ∞ ] log [2] ( 3x )
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Calculator solution
​Type in: ​lim [ x = ∞ ] ( log [2] ( 3x - 5 ) + 2 )
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Trigonometric Functions
Since most trigonometric functions are periodic (oscillating), limits at infinity do not exist except when the trigonometric expressions are part of an algebraic expression.

Illustrative Examples
Calculate each limit at infinity below.
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More Examples
Calculate each limit at infinity if it exists.
Calculator solution
Type in: lim [ x = ∞ ] ( ( 2x ) / cos ( 1 / x ) )
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Calculator solution
​Type in: ​lim [ x = ∞ ] ( x sin ( 1 / x ) )
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Functions with No Limit at x = c
Some functions oscillate as x increases such that the limit at infinity does not exist.

Examples
Calculator solution
​Type in: ​lim [ x = ∞ ] ( x / cos ( -3x ) )
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Calculator solution
​Type in: ​lim [ x = ∞ ] cos ( 2x )
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Calculator solution
​Type in: ​lim [ x = -∞ ] log ( 2x - 3 )
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Calculator solution
​Type in: ​lim [ x = -∞ ] log [2] ( 3x )
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next: 15.12. indeterminate forms >
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