Sequences of Real Numbers. Convergence
Gustavo da Silva Araújo, L. Bernal-González, José L. Gámez Merino, María E. Martínez Gómez, Gustavo A. Muñoz Fernández, Daniel L. Rodríguez Vidanes, Juan B. Seoane Sepúlveda · 2024
Use the definition of limit to verify that the limit is the one indicated for the below sequences: https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ ( 2 + 1 n + 1 ) = 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_1.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ 1 n 2 + 1 = 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_2.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ 2 n n + 1 = 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_3.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ 3 n + 1 2 n + 5 = 3 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_4.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ α + n β + n = 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_5.tif "/> , with https://www.w3.org/1998/Math/MathML" display="inline"> α , β ∈ R https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_6.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ n + 3 n 3 + 4 = 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_7.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ n 2 − 1 2 n 2 + 3 = 1 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_8.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ n 2 + 2 n + 2 n 2 + n = 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_9.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ n 2 n + n + 1 = 1 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_10.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ 1 n sin n π 4 = 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_11.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ sin n n = 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_12.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ ( n 2 + n − n ) = 1 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_13.tif "/> , https://www.w3.org/1998/Math/MathML" display="inline"> lim n → ∞ ( n 2 + 1 8 − n + 1 4 ) = 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003400745/6371057a-2ee2-4033-8adf-f9f05b7b1257/content/math3_14.tif "/> .