Let `f(x) = (xe)^(1/|x|+1/x); x != 0, f(0) = 0`, test the continuity & differentiability at `x = 0`A. discontinuous everywhereB. continuous as well as differential for all xC. continuous for all c but not differential at x=0D. neither differential nor continuous at x=0

Let `f(x) = (xe)^(1/|x|+1/x); x != 0, f(0) = 0`, test the continuity &

1.	Let `f(x) = (xe)^(1/\|x\|+1/x); x != 0, f(0) = 0`, test the continuity & differentiability at `x = 0`A. discontinuous everywhereB. continuous as well as differential for all xC. continuous for all c but not differential at x=0D. neither differential nor continuous at x=0
Answer» Correct Answer - C We have `f(x)={{:(,xe^-(-(1)/(x)+(1)/(x))=x,x lt 0),(,xe^(-2//x),x gt 0),(,0,x=0):}` `"Clearly", underset(x to 0^(-))lim f(x)=underset(x to 0)lim x=0` `underset(x to 0^(+))lim f(x)=underset(x to 0)lim xe^(-2//x)=0xx0=0 and, f(0)=0` `therefore underset(x to 0^(-))lim f(x)=underset(x to 0^(+))lim f(x)=f(0)` So, f(x) is continuous at x=0 `"Also", ("LHD at x=0")=((d)/(dx)(x))_("at x=0")=1` `("RHD at x=0")={2^(-2//x)+(2e^(-2//x))/(x)}_("at x=0")`, which does not exist Thus, f(x) is everywhere continuous but not differerntiabl at x=0