Let `f(x)={{:(cos[x]", "xle0),(|x|+a", "xlt0):}.` Then find the value of a, so that `lim_(xto0) f(x)` exists, where [x] denotes the greatest integer function less than or equal to x.

Explanation

Since `underset(xto0)limf(x)` exists, we have
`underset(xto0-)limf(x)=underset(xto0+)limf(x)`
or`""underset(hto0)limf(0-h)=underset(hto0)limf(0+h)`
or`" "underset(hto0)lim||0-h||+a=underset(hto0)cos[0+h]`
or`" "a=costheta=1`
`:." "a=1`