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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. Other 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. Scatter Plot 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

15.12. Indeterminate Forms

Indeterminate forms are undefined expressions that include: 0/0, a/0, + ∞/0,  +∞/+∞, 0( +∞), 1^∞, and ∞^0. They may result from direct substitute when we calculate the limit of a rational function as x approaches c, but it does not mean that the limit doesn't exist.

One way to resolve an indeterminate form is to simplify the given rational expression by factoring the numerator and denominator and canceling common factors. Then we plug c into the reduced rational expression.

Example
Find the limit of f(x) = (x - 1) / (x^2 - 1) as x approaches 1.
       Step 1: Check if you can use direct substitution.
                    - If you plug 1 into f(x), the result is 0/0, so you have to simplify
       Step 2: Factor the denominator (x^2 - 1) into (x - 1)(x + 1)
       Step 3: Simplify f(x) = (x - 1) / (x - 1)(x + 1) into f(x) = 1/(x + 1)
       Step 4: Plug in 1
                     The limit is f(x) = 1/(1 + 1) = 1/2
L'Hôpital's Rule
You can also use L'Hôpital's rule to find the limit of a rational function that results in an indeterminate form. The rule says that the limit of a rational function is equal to the limit of the same function after taking the derivative of the numerator and denominator respectively.
Example
Find the limit of f(x) = (x - 1) / (x^2 - 1) as x approaches 1.
​       Step 1: Take the derivative of the numerator
                    (x - 1)' = 1
       Step 2: Take the derivative of the denominator
                     (x^2 - 1)' = 2x
       Step 3: Plug 1 into the limit of f(x) = 1/2x as x approaches 1
                     f(1) = 1/(2*1) = 1/2

Calculator solution
Type in: lim[x = 1] ((x - 1) / (x^2 - 1))
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Case 1: The Limit Exists
To find a limit, you should start by trying to plug c into f(x). If you get an indeterminate form, try something else.
      1) Factor the expression.
      2) Use L'Hôpital's rule.
      3) Test points around c to determine whether the one-sided limits are in the same direction.
      4) Graph the function.

Illustrative Example
Calculate the limit.
Solution
1)  Factor the numerator and denominator.
     Factor (x - 1)^2 into (x - 1)(x - 1).
     Factor (x^2 - 1) into (x - 1)(x + 1).
 
2) Cancel out common factors.
    The simplified function is f(x) = (x - 1)/(x + 1)

3) Plug 1 into f(x) = (x - 1)/(x + 1)
    (x - 1)/( x + 1) = (1 - 1) / (1 + 1) = 0

4) The limit of (x - 1)^2 / (x^2 - 1) as x approaches 1 is 0.

Calculator solution
Type in: lim [ x = 1 ] ( ( x - 1 )^2 / ( x^2 -1 ) )
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More Examples
Calculate the limit of each rational function if it exists.
Calculator solution
Type in: lim [ x = 1 ] ( ( x - 1 ) / ( x^2 - 1 ) )
Picture
Calculator solution
​Type in: lim [ x = 2 ] ( ( x^2 - 4 ) / ( x - 2 ) )
Picture
Calculator solution
​Type in: lim [ x = 1 ] ( ( x^3 - 1 ) / ( x^2 - 1 ) )
Picture
Case 2: The Limit Does Not Exist
The limit of a function may not exist if the denominator is zero when we plug in x = c even after factoring and applying L'Hôpital's rule.
Examples
​Find each limit if it exists.
Step 1: Plug in 1
​             lim (1^2 - 1)/((1 - 1)^2) = 0/0        indeterminate form
Step 2: Use L'Hôpital's rule
              Take the derivative of the numerator       (x^2 - 1)' = 2x
              Take the derivative of the denominator   ((x - 1)^2)' = 2x - 2
Step 3: Find the left- and right-hand limits of (2x/(2x - 2))
              a) Left-hand limit
                   Plug in 0.9999
                   (2*0.9999)/((2*0.9999) - 2) ≈ 2/(-0.0002) ≈ -∞ since the denominator is almost zero but is negative
              b) Right-hand limit
                   Plug in 1.0001
                   
(2*1.0001)/((2*1.0001) - 2) ≈ 2/(0.0002) ≈ ∞ since the denominator is almost zero but is positive
Step 4: The left- and right-hand limits are not in the same direction (one goes to negative infinity and the other to positive infinity), so               the limit does not exist.
Calculator solution
Type in: lim [ x = 1 ] ( ( x^2 - 1 ) / ( x - 1 )^2 )
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Calculator solution
Type in: lim [ x=2 ] ( ( x^2 + 4x + 4 ) / ( x-2 ) )
Picture
Calculator solution
Type in: lim [ x = -2 ] ( ( x^2 - 3x + 2 ) / ( x+2 ) )
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next: 15.13. limit of a hyperbolic function >
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