ABSTRACT
If https://www.w3.org/1998/Math/MathML"> lim x → a f ( x ) = A < ∞ $ \mathop {\lim }\limits_{x \to a} f(x) = A < \infty $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> lim x → a g ( x ) = B < ∞ $ \mathop {\lim }\limits_{x \to a} g(x) = B < \infty $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> then
https://www.w3.org/1998/Math/MathML"> lim x → a ( f ( x ) ± g ( x ) ) = A ± B $ \mathop {\lim }\limits_{x \to a} (f(x) \pm g(x)) = A \pm B $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
https://www.w3.org/1998/Math/MathML"> lim x → a f ( x ) g ( x ) = A B $ \mathop {\lim }\limits_{x \to a} f(x)g(x) = AB $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
https://www.w3.org/1998/Math/MathML"> lim x → a f ( x ) g ( x ) = A B $ \mathop {\lim }\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{A}{B} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ( i f B ≠ 0 ) $ (if B \ne 0) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
https://www.w3.org/1998/Math/MathML"> lim x → a [ f ( x ) ] g ( x ) = A B $ \mathop {\lim }\limits_{x \to a} [f(x)]^{g(x)} = A^{B} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ( i f A > 0 ) $ (if A> 0) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
https://www.w3.org/1998/Math/MathML"> lim x → a h ( f ( x ) ) = h ( A ) $ \mathop {\lim }\limits_{x \to a} h(f(x)) = h(A) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (if h continuous)
If f(x) ≤ g(x) , then A ≤ B
If A = B and f(x) ≤ h(x) ≤ g(x) , then https://www.w3.org/1998/Math/MathML"> lim x → a h ( x ) = A $ \mathop {\lim }\limits_{x \to a} h(x) = A $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315154978/d4fadf3f-0d14-435f-8ef5-075606867872/content/inline-math5_10.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
