Limit at Negative Infinity of a Rational Function
Solving a limit approaching negative infinity for a rational function
Limits at Infinity of a Rational Function
Author: ShaNisaa RaSun
The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Answer" to reveal the answer to the problem.
Problem: Find \(\displaystyle \lim_{x\to \infty} f(x)\) and \(\displaystyle \lim_{x\to -\infty} f(x)\) for the following function. If a limit does not exist because the function has infinite behavior, use limit notation to describe the infinite behavior.
\[
f(x)=\dfrac{-4x^3-12x^2+15}{16x^3+14x-17}
\]
\(\displaystyle \lim_{x\to \infty} \dfrac{-4x^3-12x^2+15}{16x^3+14x-17}=-\dfrac{1}{4}\)
\(\displaystyle \lim_{x\to -\infty} \dfrac{-4x^3-12x^2+15}{16x^3+14x-17}=-\dfrac{1}{4}\)
\(\displaystyle \lim_{x\to -\infty} \dfrac{-4x^3-12x^2+15}{16x^3+14x-17}=-\dfrac{1}{4}\)
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Monotone Sequence Theorem: MATH 172 Problems 6-8
Using the Monotone Sequence Theorem and induction to show a sequence converges
Taylor and Maclaurin Series: MATH 172 Problems 1-5
Reviewing Taylor and Maclaurin Series and Taylor's Inequality
Infinite Series: MATH 172 Problems 1-3
The definition of convergent series and the Divergence Test
The Sum of a Series
Defining the sum of a series as the limit of the sequence of partial sums
Test for Divergence
Explaining how to use the Test for Divergence for an infinite series
Finding the Sum of a Series
Finding the sum of a series when you know the sequence of partial sums
Alternating Series and Ratio Tests: MATH 172 Problems 8-10
Proving the Alternating Series and Ratio Test and covers absolute convergence
Power Series: MATH 172 Problems 1-3
Covering Radius and Interval of Convergence of Power Series
Integration Using Partial Fractions: MATH 172 Problems 8 & 9
Finding a partial fraction decomposition and integrating using partial fractions
Cases for Partial Fraction Decomposition
Explaining all four cases of partial fraction decomposition
Improper Integrals: MATH 172 Problems 3-5
Definition of Improper Integrals and using the Comparison Theorem to determine convergence or divergence
Hydrostatic Force: MATH 172 Problems 6 & 7
Using Riemann Sums and Integrals to calculate hydrostatic force
WIR2 20B M251 V6
Determining the domain of a three-dimensional vector function
MATH 152: Sequences Exercise 4
Finding the limit of a sequence
MATH 152: Sequences Exercise 5
Finding the limit of a sequence with an arcsin
MATH 152: Sequences Exercise 6
Finding the limit of a sequence using logarithm properties
MATH 152: Sequences Exercise 7
Finding the limit of a sequence
WIR 20B M152 V50
Determining the limit of a bounded and increasing sequence
WIR 20B M152 V45
Determining if a sequence converges or diverges
WIR 20B M152 V46
Determining if a sequence converges or diverges
WIR 20B M152 V47
Determining if a sequence converges or diverges
WIR 20B M152 V48
Determining if a sequence is increasing, decreasing, or not monotonic and if it is bounded
WIR 20B M152 V49
Determining if a sequence is increasing, decreasing, or not monotonic and if it is bounded
Multivariable Limits Conceptual V1
Explaining how to calculate the limit of a surface for two dimensional functions
Multivariable Limits Conceptual V2
Explaining the definition of the limit of a function of two variables
Multivariable Limits Conceptual V3
Explaining limits of functions of two variables that approach infinity
Multivariable Limits Exercise 1
Solving several limits of functions of two variables
Multivariable Limits Exercise 2
Finding the limit of a function of two variables
Multivariable Limits Exercise 3
Solving a limit of a function of two variables
Multivariable Limits Exercise 4
Solving a limit of a function of two variables
Cases for Partial Fraction Decomposition
Explaining all four cases of partial fraction decomposition
Multivariable Limits Conceptual V1
Explaining how to calculate the limit of a surface for two dimensional functions
Multivariable Limits Conceptual V2
Explaining the definition of the limit of a function of two variables
Multivariable Limits Conceptual V3
Explaining limits of functions of two variables that approach infinity
WIR2 20B M150 V7
Solving a rational equation and checking the solutions
Math 140 Exam 3 Review Exercise 5
Finding the domain, intercepts, holes, and vertical asymptotes for a rational function
Math 140 Exam 3 Review Exercise 7
Compute and simplify the difference quotient for a rational function
Simplifying a Rational Expression
Simplifying a rational expression
Subtracting Rational Expressions
Adding and then simplifying two rational expressions
Multiplying Rational Expressions
Multiplying two rational expressions and then simplifying
Dividing Rational Expressions
Dividing rational expressions and then simplifying the answer
Adding Rational Expressions
Adding two rational expressions and then simplifying the answer
Holes and Asymptotes of Rational Functions
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Holes and Asymptotes of Rational Functions (2)
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Matching a Rational Function with its Graph
Matching a rational function with its graph
Domain of a Rational Function
Finding the domain of a rational function
Domain of a Rational Function (2)
Finding the domain of a rational function
Function Composition with a Rational Function
Finding the composition of a rational function and a quadratic
Inverse of a Rational Function
Finding the inverse of a rational function and its domain
Inverse of a Rational Function (2)
Finding the inverse of a rational function and its domain
Difference Quotient of a Rational Function
Finding and simplifying the difference quotient for a rational function
Difference Quotient of a Square Root Function
Finding and simplifying the difference quotient for a square root function
Solving an Inequality with a Rational Function
Solving an inequality with a rational expression by factoring and using a sign diagram
Solving an Inequality with a Sum of Rational Functions
Solving an inequality with two rational expressions using a sign diagram
Solving an Inequality with by Finding a Least Common Denominator
Solving an inequality with a rational expression using a sign diagram
A Composition of Tangent and Arccosine
Evaluating a composition of tangent and arccosine
Convergence in Probability
Defining convergence for random variables in probability
Convergence in Probability Exercise 1
Proving that a sequence of random variables converges in probability
The Sum of a Series
Defining the sum of a series as the limit of the sequence of partial sums
Test for Divergence
Explaining how to use the Test for Divergence for an infinite series
Finding the Sum of a Series
Finding the sum of a series when you know the sequence of partial sums
Math 152: Series Exercise 2
Finding the first four terms of several given series
Math 152: Series Exercise 3
Determining if a series diverges by the Test for Divergence
Math 152: Series Exercise 5
Finding the sum of a series given a formula for the partial sums
WIR 20B M152 V51
Determining if a series converges or diverges
WIR 20B M152 V59
Using the limit comparison test to determine if a series converges or diverges
WIR 20B M152 V60
Using the limit comparison test to determine if a series converges or diverges
WIR 20B M152 V61
Using the limit comparison test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 8
Using the Limit Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 9
Using the Limit Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 10
Using the Limit Comparison Test to determine if a series converges or diverges
WIR 20B M152 V62
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V63
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V64
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V65
Determining if an Alternating Series converges or diverges
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
List Comprehension and Numerical, Graphical, and Symbolic Limits in Python
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
Generating and Plotting Partials Sums of a Series in Python
Using Python to calculate the partial sums of a series to estimate the value of the series
Generating and Plotting Sequences in Python
Using Python to numerically, graphically, and analytically find the limit of a sequence
Domain of Rational Functions Video 1
Finding the domain of several rational functions
Domain of Rational Functions Video 2
Finding the domain of a rational function
Domain of Rational Functions Video 3
Explaining how to find any holes and vertical asymptotes of a rational function
Domain of Rational Functions Video 4
Finding the domain of a rational function using factoring
Domain of Combinations of Functions Video 1
Finding the domain of a function with a denominator, square root, and logarithm
Properties of Rational Functions and Expressions Video 1
Discussing parent functions of rational functions and their properties
Properties of Rational Functions and Expressions Video 2
Finding horizontal asymptotes for rational functions
Properties of Rational Functions and Expressions Video 4
Finding horizontal asymptotes for rational functions
Properties of Rational Functions and Expressions Video 5
Determining the end behavior and horizontal asymptotes for rational functions
Properties of Rational Functions and Expressions Video 7
How to simplify rational expressions
Properties of Rational Functions and Expressions Video 8
Adding and subtracting rational expressions by finding a common denominator
Properties of Rational Functions and Expressions Video 9
Explaining how to multiply and divide rational expressions
Properties of Rational Functions and Expressions Video 10
Performing algebraic operations with rational expressions and simplifying the answer
Properties of Rational Functions and Expressions Video 3
Explaining how to do polynomial long division
Solving Rational Equations Video 1
Solving equations that contain rational expressions
Solving Rational Equations Video 2
Solving an equation that contains rational expressions
Solving Rational Equations Video 3
Solving an equation that contains rational expressions
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Introduction to Riemann Sums
Estimating the area under a function using right-hand, left-hand, and midpoint Riemann sums