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: