Question

Find the value of k so that f(x) is continuous for all positive numbers?

Answer 1

`k= (1-e)/2`

Explanation:

The function is continuous for `x in (0,e)` where `f(x) = x^2-lnx+2k` and in `(e,+oo)` where `f(x) = x^2-x`.

Evaluate separately the limit of `f(x)` for `x->e` from the left and from the right:

`lim_(x->e^-) f(x) = lim_(x->e^-) x^2-lnx+2k = e^2-1+2k`

`lim_(x->e^+) f(x) = lim_(x->e^+) x^2-x = e^2-e`

For the limit `lim_(x->e) f(x)` to exist the limits from the right and from the left must concide:

`e^2-1+2k = e^2-e`

`2k = 1-e`

`k= (1-e)/2`

and in such case:

`lim_(x->e) f(x) = e^2-e = e^2 -lne +2(1-e)/2 = f(e)`

so that the function is continuous.