Question

Is the function given by f(x)= x+1/x^2-8x+12 continuous at x=2? why or why not?

Answer 1

See the explanation below

Explanation:

We have to look for the existence of a limit

`lim_(x->2)f(x)=lim_(x->2)(x+1)/(x^2-8x+12)=lim_(x->2)3/0=+oo`

So, there is a discontinuity at `x=2`

We look for the following limits

Limit on the left

`lim_(x->2^-)f(x)=lim_(x->2^-)(x+1)/(x^2-8x+12)=lim_(x->2)3/0^+=+oo`

Limit on the right

`lim_(x->2^+)f(x)=lim_(x->2^+)(x+1)/(x^2-8x+12)=lim_(x->2)3/0^-=-oo`

As `lim_(x->2^-)f(x)!=lim_(x->2^+)f(x)`, we conclude that `f(x)` is not

continuous at `x=2`