Question

Let A⊆R, let f:A⟶R and let c∈R be a cluster point of A. If lim┬(x⟶c)⁡f exists, and if |f| denotes the function defined for x∈A by |f|(x)≔|f(x)|, prove that |lim┬(x⟶c) |⁡f |=|lim┬(x⟶c)⁡|f|

Answer 1

Answer with Step-by-step explanation:

We are given that

$A\subseteq R$

Let f:$A\rightarrow R$

$c\in R$ be  a cluster point of A.

$\lim_{(x\rightarrow c)}f(x)$ exist let

$\lim_{(x\rightarrow c)}f(x)=L$

If $\mid f\mid$ denotes the function and

$\mid f(x)\mid=\mid f\mid (x)$ for $x\in A$

We have to prove that $\mid \lim_(x\rightarrow c)f(x)\mid=\lim_(x\rightarrow c)\mid f\mid (x)$

$\mid \lim_{(x\rightarrow c)}f(x)\mid=\lim_{(x\rightarrow c)}\mid f(x)\mid$

Substitute the value then we get

$\mid \lim_{(x\rightarrow c)}f(x)\mid=\mid L\mid =\lim_{(x\rightarrow c)}\mid f\mid (x)$

Hence proved.