Question

Suppose `f(x)={{:(a+bx,xlt1),(4,x=1),(b-ax,xgt1):}`
and if `lim_(xrarr1) f(x)=f(1)` what are possible values of a and b ?

Answer 1

`f(x)={{:(a+bx,xlt1),(4,x=1),(b-ax,xgt1):}`
at x=0
LHL ` =underset(xrarr0^(-))"lim"f(x)`
`underset(hrarr0)"lim"f(0-h)`
`=underset(hrarr0)"lim"(h)/(|-h|)`
`=underset(hrarr0)"lim"(h)/(-h)=underset(hrarr0)"lim"(-1)=-1`
Let ` 0-h=x`
`rArr 0-hrarr0`
`rArr hrarr0`
RHL`=underset(xrarr0^(+))"lim"f(x)`
`=underset(hrarr0)"lim"f(0+h)`
`=underset(hrarr0)"lim"( h)/(|h|)` ltbr. `=underset(hrarr0)"lim"(h)/(h)= underset(hrarr0)"lim"(1)=1`
`because LHLneRHL`
`therefore underset(xrarr0)"lim"f(x)` does not exist
Let `0+h=x`
`rArr0+hrarr0`
`rArr hrarr0`