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 $\underset{x\to c}{lim}f(x)$ and $\underset{x\to c}{lim}g(x)$ 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:

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

An equivalent formulation:

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

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 $\underset{x\to c}{lim}f(x)$ exists. Then:

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

An equivalent formulation:

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

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.

$\underset{x\to c}{lim}[a_1{f}_1(x)+a_2{f}_2(x)+\dots +a_n{f}_n(x)]=a_1\underset{x\to c}{lim}{f}_1(x)+a_2\underset{x\to c}{lim}{f}_2(x)+\dots +a_n\underset{x\to c}{lim}{f}_n(x)$

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:

• $\underset{x\to c^{-}}{lim}[f(x)+g(x)]=\underset{x\to c^{-}}{lim}f(x)+\underset{x\to c^{-}}{lim}g(x)$
• $\underset{x\to c^{+}}{lim}[f(x)+g(x)]=\underset{x\to c^{+}}{lim}f(x)+\underset{x\to c^{+}}{lim}g(x)$
• $\underset{x\to c^{-}}{lim}\lambda f(x)=\lambda \underset{x\to c^{-}}{lim}f(x)$
• $\underset{x\to c^{+}}{lim}\lambda f(x)=\lambda \underset{x\to c^{+}}{lim}f(x)$
• $\underset{x\to c^{-}}{lim}[a_1{f}_1(x)+a_2{f}_2(x)+\dots +a_n{f}_n(x)]=a_1\underset{x\to c^{-}}{lim}{f}_1(x)+a_2\underset{x\to c^{-}}{lim}{f}_2(x)+\dots +a_n\underset{x\to c^{-}}{lim}{f}_n(x)$
• $\underset{x\to c^{+}}{lim}[a_1{f}_1(x)+a_2{f}_2(x)+\dots +a_n{f}_n(x)]=a_1\underset{x\to c^{+}}{lim}{f}_1(x)+a_2\underset{x\to c^{+}}{lim}{f}_2(x)+\dots +a_n\underset{x\to c^{+}}{lim}{f}_n(x)$