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2.5: Limits at Infinity
https://math.libretexts.org/Courses/Chabot_College/MTH_1%3A_Calculus_I/02%3A_Limits/2.05%3A_Limits_at_Infinity
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f ...We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x→±∞ . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f.
5.6: Rational Functions
https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_(OpenStax)/05%3A_Polynomial_and_Rational_Functions/5.06%3A_Rational_Functions
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
4.6: Introduction to Rational Functions
https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/04%3A_Polynomial_and_Rational_Functions/4.06%3A_Introduction_to_Rational_Functions
The way we symbolize the relationship between the end behavior of $y=g(x)$ with that of the line $y=x-1$ is to write 'as $x \rightarrow \pm \infty$ , $g(x) \rightarrow x-1$ .' In this case, we s...The way we symbolize the relationship between the end behavior of $y=g(x)$ with that of the line $y=x-1$ is to write 'as $x \rightarrow \pm \infty$ , $g(x) \rightarrow x-1$ .' In this case, we say the line $y=x-1$ is a slant asymptote of $y=g(x)$ .
4.6: Rational Functions
https://math.libretexts.org/Courses/Fresno_City_College/Math_3A%3A_College_Algebra_-_Fresno_City_College/04%3A_Polynomial_and_Rational_Functions/4.06%3A_Rational_Functions
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
3.7: Rational Functions
https://math.libretexts.org/Courses/Quinebaug_Valley_Community_College/MAT186%3A_Pre-calculus_-_Walsh/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Rational_Functions
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
3.6: Introduction to Rational Functions
https://math.libretexts.org/Courses/Highline_College/MATH_141%3A_Precalculus_I_(2nd_Edition)/03%3A_Polynomial_and_Rational_Functions/3.06%3A_Introduction_to_Rational_Functions
The way we symbolize the relationship between the end behavior of $y=g(x)$ with that of the line $y=x-1$ is to write 'as $x \rightarrow \pm \infty$ , $g(x) \rightarrow x-1$ .' In this case, we s...The way we symbolize the relationship between the end behavior of $y=g(x)$ with that of the line $y=x-1$ is to write 'as $x \rightarrow \pm \infty$ , $g(x) \rightarrow x-1$ .' In this case, we say the line $y=x-1$ is a slant asymptote of $y=g(x)$ .
3.7: Rational Functions
https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Rational_Functions
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
4.5: Limits at Infinity and Asymptotes
https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_I_(Reed)/04%3A_Applications_of_Derivatives/4.05%3A_Limits_at_Infinity_and_Asymptotes
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f ...We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x→±∞ . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f.
4.4: An Interlude for Limits - L’Hospital’s Rule and Indeterminate Forms
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/04%3A_Appropriate_Applications/4.04%3A_An_Interlude_for_Limits_-_LHospitals_Rule_and_Indeterminate_Forms
This section introduces L'Hôpital's Rule, a technique for evaluating limits that result in indeterminate forms such as $0/0$ or $\infty/\infty$ . It explains how to apply the rule by differentiatin...This section introduces L'Hôpital's Rule, a technique for evaluating limits that result in indeterminate forms such as $0/0$ or $\infty/\infty$ . It explains how to apply the rule by differentiating the numerator and denominator until a determinate form is reached. The section also covers various indeterminate forms and provides examples to illustrate the use of L'Hôpital's Rule in solving complex limits.
4.5: Graphing
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT301_Calculus_I/04%3A_Applications_of_Derivatives/4.05%3A_Graphing
The second derivative is zero at $x=0.$ Therefore, to determine the concavity of $f$ , divide the interval $(−∞,∞)$ into the smaller intervals $(−∞,0)$ and $(0,∞)$ , and choose test points \(x...The second derivative is zero at $x=0.$ Therefore, to determine the concavity of $f$ , divide the interval $(−∞,∞)$ into the smaller intervals $(−∞,0)$ and $(0,∞)$ , and choose test points $x=−1$ and $x=1$ to determine the concavity of $f$ on each of these smaller intervals as shown in the following table.
3.7: Rational Functions
https://math.libretexts.org/Courses/Hartnell_College/MATH_25%3A_PreCalculus_(Abramson_OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Rational_Functions
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

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