Limit is linear

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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 \lambda 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} (\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_1f_1(x) + a_2f_2(x) + \dots + a_nf_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_1f_1(x) + a_2f_2(x) + \dots + a_nf_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_1f_1(x) + a_2f_2(x) + \dots + a_nf_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)