New Approaches for Evaluation Indeterminate Limits for Multivariable Functions in Undergraduate Mathematics Courses

Çiğdem Dinçkal

Makale detayları

New Approaches for Evaluation Indeterminate Limits for Multivariable Functions in Undergraduate Mathematics Courses

Yazarlar:

Çiğdem Dinçkal

Yayıncı:

Gaziantep University

Ana Başlıklar:

Applied Mathematics (Other)

Dergi:

Natural Sciences and Engineering Bulletin Vol.2 Issue 1, pp.56-74

Belge Tipi:

Makale

Yayın Tarihi:

5/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001349310

pISSN:

3023-8293

Anahtar kelimeler:

Özet:

 Zero divided by zero is one of the most important indeterminate forms obtained when evaluating limits for single variable functions and series in calculus education. Well-known method; LHôpital rule and its generalized form have been employed to simplify and resolve the indeterminate form such that zero divided by zero in terms of quotients of their derivatives for single variable functions as well as for multivariable functions. Nevertheless, L Hôpital rule is impractical for the indeterminate limit forms of two variable functions in some cases such that isolated and nonisolated singularities, requirement of application of LHôpital rule more than once and complexity of taking derivative for some multivariable functions. So LHôpital rule cannot be preferred due to these reasons. By considering all these facts, new approaches including Finite Differences such as Central CFD, Forward FFD, Backward BFD, High Accurate Central HACFD, High Accurate Forward HAFFD, High Accurate Backward HABFD methods, and Richardson Extrapolation method are presented that provide efficient ways to solve these limits instead of using L Hôpital rule. Error analysis is also performed. All these methods are compared with each other in terms of accuracy and computational efficiency. It is observed that these approaches will be good alternatives instead of LHôpital rule for indeterminate form of two variable functions in calculus courses for both instructors and their students. Numerical examples are presented for this purpose.

 Zero divided by zero is one of the most important indeterminate forms obtained when evaluating limits for single variable functions and series in calculus education. Well-known method; LHôpital rule and its generalized form have been employed to simplify and resolve the indeterminate form such that zero divided by zero in terms of quotients of their derivatives for single variable functions as well as for multivariable functions. Nevertheless, L Hôpital rule is impractical for the indeterminate limit forms of two variable functions in some cases such that isolated and nonisolated singularities, requirement of application of LHôpital rule more than once and complexity of taking derivative for some multivariable functions. So LHôpital rule cannot be preferred due to these reasons. By considering all these facts, new approaches including Finite Differences such as Central CFD, Forward FFD, Backward BFD, High Accurate Central HACFD, High Accurate Forward HAFFD, High Accurate Backward HABFD methods, and Richardson Extrapolation method are presented that provide efficient ways to solve these limits instead of using L Hôpital rule. Error analysis is also performed. All these methods are compared with each other in terms of accuracy and computational efficiency. It is observed that these approaches will be good alternatives instead of LHôpital rule for indeterminate form of two variable functions in calculus courses for both instructors and their students. Numerical examples are presented for this purpose.

 Zero divided by zero is one of the most important indeterminate forms obtained when evaluating limits for single variable functions and series in calculus education. Well-known method; LHôpital rule and its generalized form have been employed to simplify and resolve the indeterminate form such that zero divided by zero in terms of quotients of their derivatives for single variable functions as well as for multivariable functions. Nevertheless, L Hôpital rule is impractical for the indeterminate limit forms of two variable functions in some cases such that isolated and nonisolated singularities, requirement of application of LHôpital rule more than once and complexity of taking derivative for some multivariable functions. So LHôpital rule cannot be preferred due to these reasons. By considering all these facts, new approaches including Finite Differences such as Central CFD, Forward FFD, Backward BFD, High Accurate Central HACFD, High Accurate Forward HAFFD, High Accurate Backward HABFD methods, and Richardson Extrapolation method are presented that provide efficient ways to solve these limits instead of using L Hôpital rule. Error analysis is also performed. All these methods are compared with each other in terms of accuracy and computational efficiency. It is observed that these approaches will be good alternatives instead of LHôpital rule for indeterminate form of two variable functions in calculus courses for both instructors and their students. Numerical examples are presented for this purpose.

 Zero divided by zero is one of the most important indeterminate forms obtained when evaluating limits for single variable functions and series in calculus education. Well-known method; LHôpital rule and its generalized form have been employed to simplify and resolve the indeterminate form such that zero divided by zero in terms of quotients of their derivatives for single variable functions as well as for multivariable functions. Nevertheless, L Hôpital rule is impractical for the indeterminate limit forms of two variable functions in some cases such that isolated and nonisolated singularities, requirement of application of LHôpital rule more than once and complexity of taking derivative for some multivariable functions. So LHôpital rule cannot be preferred due to these reasons. By considering all these facts, new approaches including Finite Differences such as Central CFD, Forward FFD, Backward BFD, High Accurate Central HACFD, High Accurate Forward HAFFD, High Accurate Backward HABFD methods, and Richardson Extrapolation method are presented that provide efficient ways to solve these limits instead of using L Hôpital rule. Error analysis is also performed. All these methods are compared with each other in terms of accuracy and computational efficiency. It is observed that these approaches will be good alternatives instead of LHôpital rule for indeterminate form of two variable functions in calculus courses for both instructors and their students. Numerical examples are presented for this purpose.

Makale detayları

New Approaches for Evaluation Indeterminate Limits for Multivariable Functions in Undergraduate Mathematics Courses

Yazarlar:

Çiğdem Dinçkal

Yayıncı:

Gaziantep University

Ana Başlıklar:

Applied Mathematics (Other)

Dergi:

Natural Sciences and Engineering Bulletin Vol.2 Issue 1, pp.56-74

Belge Tipi:

Makale

Yayın Tarihi:

5/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001349310

pISSN:

3023-8293

Zero divided by zero

L’Hôpital rule

Central finite differences

Forward finite differences

Backward finite differences

Richardson extrapolation method

Sayı

Makale detayları

New Approaches for Evaluation Indeterminate Limits for Multivariable Functions in Undergraduate Mathematics Courses

Yazarlar: Çiğdem Dinçkal

Yayıncı: Gaziantep University

Ana Başlıklar: Applied Mathematics (Other)

Dergi: Natural Sciences and Engineering Bulletin Vol.2 Issue 1, pp.56-74

Belge Tipi: Makale

Yayın Tarihi: 5/2025

Belge Dili: İngilizce

Document ID (Org No.): 2001349310

pISSN: 3023-8293

Zero divided by zero

L’Hôpital rule

Central finite differences

Forward finite differences

Backward finite differences

Richardson extrapolation method

Sayı

Yazarlar:

Çiğdem Dinçkal

Yayıncı:

Gaziantep University

Ana Başlıklar:

Applied Mathematics (Other)

Dergi:

Natural Sciences and Engineering Bulletin Vol.2 Issue 1, pp.56-74

Belge Tipi:

Makale

Yayın Tarihi:

5/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001349310

pISSN:

3023-8293