Limit is linear

Statement

In terms of additivity and pulling out scalars

Additive:

Suppose $f$ and $g$ are functions of one variable. Suppose $c\in \mathbb {R}$ is such that both $f$ and $g$ are defined on the immediate left and the immediate right of $c$. Further, suppose that the limits $\lim _{x\to c}f(x)$ and $\lim _{x\to c}g(x)$ both exist (as finite numbers). In that case, the limit of the pointwise sum of functions $f+g$ exists and is the sum of the individual limits:

$\lim _{x\to c}(f+g)(x)=\lim _{x\to c}f(x)+\lim _{x\to c}g(x)$

An equivalent formulation:

$\!\lim _{x\to c}[f(x)+g(x)]=\lim _{x\to c}f(x)+\lim _{x\to c}g(x)$

Scalars: Suppose $f$ is a function of one variable and $\lambda$ is a real number. Suppose $c\in \mathbb {R}$ is such that $f$ is defined on the immediate left and immediate right of $c$, and that $\lim _{x\to c}f(x)$ exists. Then:

$\lim _{x\to c}(\lambda f)(x)=\lambda \lim _{x\to c}f(x)$

An equivalent formulation:

$\lim _{x\to c}\lambda f(x)=\lambda \lim _{x\to c}f(x)$

In terms of generalized linearity

Suppose $f_{1},f_{2},\dots ,f_{n}$ are functions and $a_{1},a_{2},\dots ,a_{n}$ are real numbers.

$\lim _{x\to c}[a_{1}f_{1}(x)+a_{2}f_{2}(x)+\dots +a_{n}f_{n}(x)]=a_{1}\lim _{x\to c}f_{1}(x)+a_{2}\lim _{x\to c}f_{2}(x)+\dots +a_{n}\lim _{x\to c}f_{n}(x)$

if the right side expression makes sense.

In particular, setting $n=2,a_{1}=1,a_{2}=-1$, we get that the limit of the difference is the difference of the limits.

One-sided version

One-sided limits (i.e., the left hand limit and the right hand limit) are also linear. In other words, we have the following, whenever the respective right side expressions make sense:

• $\!\lim _{x\to c^{-}}[f(x)+g(x)]=\lim _{x\to c^{-}}f(x)+\lim _{x\to c^{-}}g(x)$
• $\!\lim _{x\to c^{+}}[f(x)+g(x)]=\lim _{x\to c^{+}}f(x)+\lim _{x\to c^{+}}g(x)$
• $\!\lim _{x\to c^{-}}\lambda f(x)=\lambda \lim _{x\to c^{-}}f(x)$
• $\!\lim _{x\to c^{+}}\lambda f(x)=\lambda \lim _{x\to c^{+}}f(x)$
• $\lim _{x\to c^{-}}[a_{1}f_{1}(x)+a_{2}f_{2}(x)+\dots +a_{n}f_{n}(x)]=a_{1}\lim _{x\to c^{-}}f_{1}(x)+a_{2}\lim _{x\to c^{-}}f_{2}(x)+\dots +a_{n}\lim _{x\to c^{-}}f_{n}(x)$
• $\lim _{x\to c^{+}}[a_{1}f_{1}(x)+a_{2}f_{2}(x)+\dots +a_{n}f_{n}(x)]=a_{1}\lim _{x\to c^{+}}f_{1}(x)+a_{2}\lim _{x\to c^{+}}f_{2}(x)+\dots +a_{n}\lim _{x\to c^{+}}f_{n}(x)$