Limit is linear

Statement

In terms of additivity and pulling out scalars

Additive:

Suppose $f$ and $g$ are functions of one variable. Suppose $c \in R$ 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 $λ$ is a real number. Suppose $c \in R$ 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:

$lim_{x \to c} (λ f) (x) = λ lim_{x \to c} f (x)$

An equivalent formulation:

$lim_{x \to c} λ f (x) = λ lim_{x \to c} f (x)$

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:

$lim_{x \to c^{-}} [f (x) + g (x)] = lim_{x \to c^{-}} f (x) + lim_{x \to c^{-}} g (x)$
$lim_{x \to c^{+}} [f (x) + g (x)] = lim_{x \to c^{+}} f (x) + lim_{x \to c^{+}} g (x)$
$lim_{x \to c^{-}} λ f (x) = λ lim_{x \to c^{-}} f (x)$
$lim_{x \to c^{+}} λ f (x) = λ lim_{x \to c^{+}} f (x)$
$lim_{x \to c^{-}} [a_{1} f_{1} (x) + a_{2} f_{2} (x) + \dots + a_{n} f_{n} (x)] = a_{1} lim_{x \to c^{-}} f_{1} (x) + a_{2} lim_{x \to c^{-}} f_{2} (x) + \dots + a_{n} lim_{x \to c^{-}} f_{n} (x)$
$lim_{x \to c^{+}} [a_{1} f_{1} (x) + a_{2} f_{2} (x) + \dots + a_{n} f_{n} (x)] = a_{1} lim_{x \to c^{+}} f_{1} (x) + a_{2} lim_{x \to c^{+}} f_{2} (x) + \dots + a_{n} lim_{x \to c^{+}} f_{n} (x)$