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.