Question

If ` f(x)={{:(|x|+1,xlt0),(0,x=0) ,(|x|-1,xgt0):}`
for what value (s) of a does `lim_(xrarra) f(x)` exists?

Answer 1

`f(x)={{:(|x|+1,xlt0),(0,x=0),(|x|-1,xgt0):}={{:(-x+1,xlt0),(0,x=0),(x-1,xgt0):}`
`xlt0` then `f(x)= -x+1` is a polynomial function
`therefore ` for each `alt0`, lim f(x) exists.
`xgt0` then `f(x)=x-1` is a polynomial function.
`therefore` For each `a gt0,underset(Xrarr1)"lim"f(x)` exists.
at x=0
LHL`=underset(xrarr0^(-))"lim"f(x)`
`underset(hrarr0)"lim"f(0-h)`
`=underset(hrarr0)"lim"-( 0-h)+1`
`=0+1=1`
Let `0-h=x`
`rArr0-hrarr0`
`rArr hrarr0`
LHL `=underset(xrarr0^(+))"lim"f(x)`
`=underset(hrarr0)"lim"f(0+h)`
`=underset(hrarr0)"lim"(0+h)-1=0-1=-1`
`because LHLneRHL`.
`therefore underset(Xrarr0)"lim"f(x)` does not exist.
therefore `underset(xrarra)"lim" f(x)` exists for each `a ne0`.