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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

15.5. Limit of a Rational Function

Limit of a Rational Function
A rational function may have a restricted value at x = c such that finding the limit is not straightforward. The rules are listed as follows:

1) Determine the restricted values for the domain of the function.
    To find these values, set the denominator to 0 and find the roots of the resulting equation.

Example
f(x) = 3/(x - 4)

When x = 4, the function is undefined, implying that the graph has an infinite discontinuity such that the limit does not exist as x approaches 4. The line x=4 is a vertical asymptote of the graph.
Picture
Picture

2) If c is a restricted value, the limit may or may not exist.To determine whether it exists, simplify the rational expression. The limit            exists if the simplified form is no longer a rational expression or the denominator is no longer zero when x = c. 

3) Plug in c after you simplify the expression. If the denominator is still zero, find the one-sided limits of the function by plugging in values that are close to c from the left and right. If the one-sided limits are equal, the limit equals the left- and right-hand limits. Otherwise the limit does not exist.

4) If c is not in the restricted domain, plug it into the rational expression as you usually would.
Finding the Limit Through Direct Substitution
In each rational function below, the value of c is not in the restricted domain so the limit can be found by plugging in the value of c.

Examples
Find the indicated limit.
Calculator solution
​Type in: lim [ x = -3 ] ( ( 2x ) / (3x - 1 ) )
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Calculator solution
​Type in: lim [ x = 2 + ] ( ( 3 - x ) / ( 3x + 1 ) )
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Calculator solution
​Type in: lim [ x = 1000000 ] ( 1 / ( x - 1 )^2 )
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Calculator solution
​Type in: lim [ x = 2 ] ( 1 / ( x - 1 )^2 )
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Limit at a Restricted Value of X
In each rational function below, the value of c is a restricted value of the function's domain. There are three cases that could happen:

Case 1: The Limit Exists
The limit exists at the restricted value if the original rational function can be simplified to cancel out the denominator. At x = c, the graph has a hole​.

Illustrative Example
Consider the function f(x) = (x^3 - 4x^2 + x + 6) / (x - 2) and find the limit as x approaches 2. The function is undefined at x = 2, but the expression can be simplified to f(x) = x^2 - 2x - 3. Because the denominator cancels out, the limit as x approaches 2 exists.

Note that finding the limit of f(x) = (x^3 - 4x^2 + x + 6) / (x - 2) gives the same result as finding the limit of ​f(x) = x^2 - 2x - 3 as shown below.
Picture
The graph below shows that the graph still has a hole at x = 2.
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More Examples
Evaluate the limit of each rational function as x approaches c where c is a restricted value of the domain.
Calculator solution
​Type in: lim [ x = 2 ] ( ( x^2 + 4x - 12 ) / ( x^2 - 2x ) )
Picture
Calculator solution
​Type in: ​lim [ t = 5 ] ( ( t^3 - 6t^2 + 25 ) / ( t - 5 ) )
Picture

Calculator solution
​Type in: ​lim [ x = 1 ] ( ( 2 - 2x^2 ) / ( x - 1 ) )
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Calculator solution
​Type in: ​lim [ x = -6 ] ( ( ( 2x + 8 ) / ( x^2 - 12 ) - 1 / x ) / ( x + 6 ) )
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Calculator solution
​Type in: ​lim [ h = 0 ] ( ( 2 ( -3 + h )^2 - 18 ) / h )
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Calculator solution
​Type in: ​lim [ x = 0 ] ( ( ( 1 / ( 3 + x ) - 1 / ( 3 - x ) ) ) / x )
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Case 2: Limit to Infinity
The limit may approach infinity if the denominator is still zero at x = c. As a result, the graph of the function continues increasing or decreasing as x approaches c from the left and right. At this value of x, the graph has an infinite discontinuity. The limit exists if the left- and right-hand limits increase or decrease in the same direction.

Illustrative Example
Consider the function f(x) = 1/x^2 and find the limit as x approaches 0. Because the denominator is zero when we plug in x = 0, we have to evaluate the function from the left and right. The left- and right-hand limits both approach positive infinity, so the limit of f(x) as x approaches 0 is ∞.
Picture
Picture
More Examples
Calculator solution
Type in: lim [ x = 0 ] ( 6 / x^2 )
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Calculator solution
Type in: lim [ x = 1 ] ( 1 / ( x - 1 )^2 )
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Calculator solution
Type in: lim [ x = 1 ] ( ( x - 2 ) / ( x - 1 )^2 )
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Calculator solution
Type in: lim [ x = 0 ] ( ( 6 - 3x ) / ( 2x^2 ) )
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Calculator solution
Type in: lim [ x = - 4 ] ( x / ( 2x + 8 )^2 )
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Case 3: The Limit Does Not Exist (DNE)
When the denominator is zero at x = c, x = c is a vertical asymptote of the graph. The function either continues increasing or decreasing as x approaches c from the left and right. If the left- and right-hand limits are in different directions, the limit does not exist.

Illustrative Example
Consider the limit of f(x) = 1/x as x approaches 0. The function is undefined at x = 0, as shown below. Because the left-hand limit is negative infinity while the right-hand limit is positive infinity, the limit as x approaches 0 does not exist.
Picture
Picture
More Examples
Calculator solution
​Type in: lim [ x = -2 ] ( -4 / ( x + 2 ) )
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Calculator solution
​Type in: ​lim [ t = 0 ] ( 1 / t - 1 / ( t^2 + 1 ) )
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Calculator solution
​Type in: ​lim [ x = 4 ] ( 3 / ( 4 - x )^3 )
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Calculator solution
​Type in: ​lim [ x = 2 ] ( 1 / ( x - 2 ) )
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Calculator solution
​Type in: ​lim [ x = 4 ] ( ( x^2 - 3x - 10 ) / ( 3x - 12 ) )
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next: 15.6. limit of a radical function >
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