# Limit is linear

## Contents

## Statement

### In terms of additivity and pulling out scalars

*Additive*:

Suppose and are functions of one variable. Suppose is such that both and are defined on the immediate left and the immediate right of . Further, suppose that the limits and both exist (as finite numbers). In that case, the limit of the pointwise sum of functions exists and is the sum of the individual limits:

An equivalent formulation:

*Scalars*: Suppose is a function of one variable and is a real number. Suppose is such that is defined on the immediate left and immediate right of , and that exists. Then:

An equivalent formulation:

### In terms of generalized linearity

Suppose are functions and are real numbers.

if the right side expression makes sense.

In particular, setting , 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: