Combination Theorem for Limits of Functions/Real/Multiple Rule
< Combination Theorem for Limits of Functions | Real(Redirected from Multiple Rule for Limits of Real Functions)
Jump to navigation
Jump to search
Theorem
Let $\R$ denote the real numbers.
Let $f$ be a real function defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ tend to the following limit:
- $\ds \lim_{x \mathop \to c} \map f x = l$
Let $\lambda \in \R$ be an arbitrary real number.
Then:
- $\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$
Proof
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
- $\forall n \in \N_{>0}: x_n \ne c$
- $\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
- $\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the Multiple Rule for Real Sequences:
- $\ds \lim_{n \mathop \to \infty} \lambda \map f {x_n} = \lambda l$
Applying Limit of Real Function by Convergent Sequences again:
- $\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$
$\blacksquare$