Question

3. Prove that f(x) =1/(x-1)
is discontinuous at x=1.

Answer 1

Answer:

One can prove the discontinuity of any function by finding out the limit of the function from both sides of the value of the independent value where it's predicted to blow up or not give a finite value.

Clearly, 1 is not in the domain of this function that is, f(x) = 1/x-1

So, straight away we can say that this function is discontinuous however to prove it rigorously

first finding limit from left hand side

So,

LHL = lim x-> 1-. 1/x-1 = 1/0- -> - infinity so it's not a finite distinct value

Similarly

RHL = Lim x->1+. 1/x-1 = 1/0+ -> + infinity

So,since both LHL and RHL are not finite as well as same, we can now say that the function given is discontinuous at x=1

Also, this is not a removable discontinuity as there is no possible finite value that we can assign other than what this function explicitly approaches.

So, it also has at x=1 what's called the non-removable discontinuity.

Hope this helps you !