nyuuzyou/wikis huggingface dataset:
Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule

== Theorem ==
Let $\struct {S, \tau_S}$ be a topological space.
Let $\struct {G, *, \tau_G}$ be a topological semigroup.
Let $\lambda \in G$.
Let $f: \struct {S, \tau_S} \to \struct {G, \tau_G}$ be a continuous mapping.
Let $\lambda * f: S \to G$ be the mapping defined by:
:$\forall x \in S: \map {\paren {\lambda * f} } x = \lambda * \map f x$
Let $f * \lambda: S \to G$ be the mapping defined by:
:$\forall x \in S: \map {\paren {f * \lambda} } x = \map f x * \lambda$
Then:
:$\lambda * f: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping
:$f * \lambda: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping.
== Proof ==
Let $c_\lambda : S \to G$ be the constant mapping defined by:
:$\forall x \in S: \map {c_\lambda} x = \lambda$
From Constant Mapping is Continuous, $c_\lambda$ is continuous.
From Product Rule for Continuous Mappings to Topological Semigroup:
:$c_\lambda * f$ and $f * c_\lambda$ are continuous.
Now:
From Equality of Mappings:
:$c_\lambda * f = \lambda * f$
Similarly:
:$f * c_\lambda = f * \lambda$
The result follows.
Category:Combination Theorem for Continuous Mappings to Topological Semigroup
Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule

== Theorem ==
Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {G, *, \tau_{_G} }$ be a topological semigroup.
Let $f, g: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ be continuous mappings.
Let $f * g: S \to G$ be the mapping defined by:
:$\forall x \in S: \map {\paren {f * g} } x = \map f x * \map g x$
Then:
:$f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
== Proof ==
Let $f \times g: S \to G \times G$ be the mapping defined by:
:$\forall x \in S : \map {\paren {f \times g} } x = \tuple {\map f x, \map g x}$
From Pointwise Operation is Composite of Operation with Mapping to Cartesian Product:
:$f * g = * \circ \paren {f \times g}$
Let $\tau$ be the product topology on $G \times G$.
From Corollary to Continuous Mapping to Product Space:
:$f \times g : \struct{S, \tau_{_S}} \to \struct {G \times G, \tau}$ is a continuous mapping.
Now $* : \struct {G \times G, \tau} \to \struct {G, \tau_{_G} }$ is continuous by definition of a topological semigroup.
From Composite of Continuous Mappings is Continuous:
:$* \circ \paren {f \times g}$ is continuous.
The result follows.
Category:Combination Theorem for Continuous Mappings to Topological Semigroup
Combination Theorem for Continuous Real-Valued Functions

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be continuous real-valued functions.
Let $\lambda \in \R$.
Then the following results hold.
=== Sum Rule ===
=== Negation Rule ===
=== Difference Rule ===
=== Product Rule ===
=== Multiple Rule ===
=== Absolute Value Rule ===
=== Maximum Rule ===
=== Minimum Rule ===
Category:Combination Theorem for Continuous Real-Valued Functions 
Category:Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Absolute Value Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f : S \to \R$ be a continuous real-valued function.
Let $\size f : S \to \R$ denote the absolute value of $f$, that is, $\size f$ denotes the mapping defined by:
:$\forall s \in S : \map {\size f} s = \size{\map f s}$
 
Then:
:$\size f$ is a continuous real-valued function
== Proof ==
From Absolute Value of Function is Composite with Absolute Value Function:
:$\size{f} = \size{\,\cdot\,} \circ f$
where:
:$\size{\,\cdot\,}$ denotes the absolute value function $\size{\,\cdot\,} : \R \to \R$
:$\size{\,\cdot\,} \circ f$ denotes the composite mapping of $\size{\,\cdot\,}$ with $f$
From Absolute Value Function is Continuous:
:$\size{\,\cdot\,} : \R \to \R$ is continuous
From Composite of Continuous Mappings is Continuous:
:$\size{\,\cdot\,} \circ f$ is continuous
The result follows.
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Difference Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be contiuous real-valued functions.
Let $f - g : S \to \R$ be the pointwise difference of $f$ and $g$, that is, $f - g$ is the mappping defined by:
:$\forall s \in S : \map {\paren{f - g} } s = \map f s - \map g s$
Then:
:$f - g$ is a coninuous real-valued function
== Proof ==
From Pointwise Difference is Pointwise Addition with Negation:
:$f - g = f + \paren{-g}$
where:
:$-g$ denotes the pointwise negation of $g$
:$f + \paren{-g}$ denotes the pointwise addition of $f$ and $-g$
From Negation Rule for Continuous Real-Valued Function:
:$-g$ is a contiuous real-valued function
From Sum Rule for Continuous Real-Valued Functions
:$f + \paren{-g}$ is a contiuous real-valued function
The result follows.
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Maximum Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be continuous real-valued functions.
Let $f \vee g$ denote the pointwise maximum of $f$ and $g$, that is, $f \vee g$ is the mapping defined by:
:$\forall s \in S : \map {\paren{f \vee g} } s = \max \set{\map f s, \map g s}$
Then:
:$f \vee g$ is a continuous real-valued function
== Proof ==
From Characterization of Pointwise Maximum of Real-Valued Functions:
::$f \vee g = \dfrac 1 2 \paren{f + g + \size{f - g}}$
We have:
The result follows.
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Minimum Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be continuous real-valued functions.
Let $f \wedge g$ denote the pointwise minimum of $f$ and $g$, that is, $f \wedge g$ is the mapping defined by:
:$\forall s \in S : \map {\paren{f \wedge g} } s = \min \set{\map f s, \map g s}$
Then:
:$f \wedge g$ is a continuous real-valued function
== Proof ==
From Characterization of Pointwise Minimum of Real-Valued Functions:
::$f \vee g = \dfrac 1 2 \paren{f + g - \size{f - g}}$
We have:
The result follows.
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Multiple Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f :S \to \R$ be contiuous real-valued function.
Let $\lambda \in \R$.
Let $\lambda f : S \to \R$ be the pointwise scalar multiplication of $f$ by $\lambda$, that is, $\lambda f$ is the mappping defined by:
:$\forall s \in S : \map {\paren{\lambda f} } s = \lambda \map f s$
Then:
:$\lambda f$ is a continuous real-valued function
== Proof ==
Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Multiple Rule for Continuous Mappings into Normed Division Ring
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Negation Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be continuous real-valued functions.
Let $-f : S \to \R$ denote the pointwise negation of $f$, that is, $-f$ denotes the mapping defined by:
:$\forall s \in S : \map {\paren{-f} } s = - \map f s$
Then:
:$-f$ is a continuous real-valued function
== Proof ==
Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Negation Rule for Continuous Mappings into Normed Division Ring
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Product Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be continuous real-valued functions.
Let $f g : S \to \R$ be the pointwise product of $f$ and $g$, that is, $f g$ is the mappping defined by:
:$\forall s \in S : \map {\paren{f g} } s = \map f s \map g s$
Then:
:$f g$ is a continuous real-valued function
== Proof ==
Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Product Rule for Continuous Mappings into Normed Division Ring
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions/Sum Rule

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be contiuous real-valued functions.
Let $f + g : S \to \R$ be the pointwise addition of $f$ and $g$, that is, $f + g$ is the mappping defined by:
:$\forall s \in S : \map {\paren{f + g} } s = \map f s + \map g s$
Then:
:$f + g$ is a coninuous real-valued function
== Proof ==
Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Sum Rule for Continuous Mappings into Normed Division Ring
Category:Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Limits of Functions/Combined Sum Rule

== Theorem ==
=== Real Functions ===
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Let $\lambda, \mu \in \R$ be arbitrary real numbers.
Then:
=== Complex Functions ===
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Let $\lambda, \mu \in \C$ be arbitrary complex numbers.
Then:
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Complex

== Theorem ==
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Let $\lambda, \mu \in \C$ be arbitrary complex numbers.
Then the following results hold:
=== Sum Rule ===
=== Multiple Rule ===
=== Combined Sum Rule ===
=== Product Rule ===
=== Quotient Rule ===
== Also see ==
* Combination Theorem for Continuous Functions
Category:Limits of Complex Functions
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Complex/Combined Sum Rule

== Theorem ==
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Let $\lambda, \mu \in \C$ be arbitrary complex numbers.
Then:
:$\ds \lim_{z \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$
== Proof ==
Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Combined Sum Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda \map f {z_n} + \mu \map g {z_n} } = \lambda l + \mu m$
Applying Limit of Complex Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Complex/Multiple Rule

== Theorem ==
Let $\C$ denote the complex numbers.
Let $f$ be a complex function defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ tend to the following limit:
:$\ds \lim_{z \mathop \to c} \map f z = l$
Let $\lambda \in \C$ be an arbitrary complex number.
Then:
:$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$
== Proof ==
Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
By the Multiple Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty}\lambda \map f {z_n} = \lambda l$
Applying Limit of Complex Function by Convergent Sequences again:
:$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Complex/Product Rule

== Theorem ==
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Then:
:$\ds \lim_{z \mathop \to c} \ \paren {\map f z \map g z} = l m$
== Proof ==
Let $\sequence {z_n}$ be a sequence of elements of $S$ such that:
:$\forall n \in \N: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Product Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\map f {z_n} \map g {z_n} } = l m$
Applying Limit of Complex Function by Convergent Sequences again, we get:
:$\ds \lim_{z \mathop \to c} \paren {\map f z \map g z} = l m$
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Complex/Quotient Rule

== Theorem ==
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Then:
:$\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$
provided that $m \ne 0$.
== Proof ==
Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Quotient Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \frac {\map f {z_n} } {\map g {z_n} } = \frac l m$
provided that $m \ne 0$.
Applying Limit of Real Function by Convergent Sequences again, we get:
:$\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$
provided that $m \ne 0$.
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Complex/Sum Rule

== Theorem ==
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Then:
:$\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$
== Proof ==
Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Sum Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\map f {z_n} + \map g {z_n} } = l + m$
Applying Limit of Complex Function by Convergent Sequences again, we get:
:$\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Multiple Rule

== Theorem ==
=== Real Functions ===
Let $\R$ denote the real numbers.
Let $f$ be a real function defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ tend to the following limit:
:$\ds \lim_{x \mathop \to c} \map f x = l$
Let $\lambda \in \R$ be an arbitrary real number.
Then:
=== Complex Functions ===
Let $\C$ denote the complex numbers.
Let $f$ be a complex function defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ tend to the following limit:
:$\ds \lim_{z \mathop \to c} \map f z = l$
Let $\lambda \in \C$ be an arbitrary complex number.
Then:
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Product Rule

== Theorem ==
=== Real Functions ===
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Then:
=== Complex Functions ===
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Then:
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Quotient Rule

== Theorem ==
=== Real Functions ===
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Then:
=== Complex Functions ===
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Then:
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Real

== Theorem ==
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Let $\lambda, \mu \in \R$ be arbitrary real numbers.
Then the following results hold:
=== Sum Rule ===
=== Multiple Rule ===
=== Combined Sum Rule ===
=== Product Rule ===
=== Quotient Rule ===
== Also see ==
* Combination Theorem for Continuous Functions
Category:Limits of Real Functions
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Real/Combined Sum Rule

== Theorem ==
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Let $\lambda, \mu \in \R$ be arbitrary real numbers.
Then:
:$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N^*: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Combined Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda \map f {x_n} + \mu \map g {x_n} } = \lambda l + \mu m$
Applying Limit of Real Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$
== Sources ==
* : $\S 8.12 \ \text{(i)}$
* : Appendix: $\S 18.6$: Limits of functions
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Real/Multiple Rule

== Theorem ==
Let $\R$ denote the real numbers.
Let $f$ be a real function defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ tend to the following limit:
:$\ds \lim_{x \mathop \to c} \map f x = l$
Let $\lambda \in \R$ be an arbitrary real number.
Then:
:$\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \lambda \map f {x_n} = \lambda l$
Applying Limit of Real Function by Convergent Sequences again:
:$\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Real/Product Rule

== Theorem ==
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Then:
:$\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$
== Proof ==
Let $\sequence {x_n}$ be a sequence of elements of $S$ such that:
:$\forall n \in \N: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Product Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\map f {x_n} \map g {x_n} } = l m$
Applying Limit of Real Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$
== Sources ==
* : Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(ii)}$
* : $1$: Review of some real analysis: $\S 1.4$: Continuity
* : $\S 8.12 \ \text{(ii)}$
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Real/Quotient Rule

== Theorem ==
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Then:
:$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$.
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Quotient Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \frac {\map f {x_n} } {\map g {x_n} } = \frac l m$
provided that $m \ne 0$.
Applying Limit of Real Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$.
== Also see ==
* L'Hôpital's Rule: for the case where $l = m = 0$
== Sources ==
* : Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(iii)}$
* : $1$: Review of some real analysis: $\S 1.4$: Continuity
* : $\S 8.12 \ \text{(iii)}$
Category:Combination Theorems for Limits of Functions