Question

Show that `lim_(xrarr2)(x)/([x])` does not exist.

Answer 1

Let `f(x)=(x)/({x}).` Then,
`lim_(xto2^(+))f(x)=lim_(hto0)f(2+h)=lim_(hto0)(2+h)/([2+h])=1" "[becausehto0,[2+h]=2]`
`lim_(xto2^(-))f(x)=lim_(hto0)f(2-h)=lim_(hto0)(2-h)/([2-h])=lim_(hto0)(2-h)/(1)=2" "[because[2-h]=1]`
`thereforelim_(xto2^(+))f(x)nelim_(xto2^(-))f(x)and solim_(xto2)f(x)` does not exist.