Question

Find the Value of k so that the function f is continuous:

f(x)= {9x^2-4/3x-2, x≠-2/3 }

k , x=-2/3

Answer 1

`k=(26)/(9)`

Explanation:

`f(x)=9x^2-(4)/(3)x-2, x\\eq (-2)/(3)` and
`f(x)=k, x=(-2)/(3)`

Remember that a function is continuous in x=a if
`lim_(x\rightarrow a)f(x)=f(a)`

Since f is a rational function for every number different of
`x=-(2)/(3)`, then f is continuous in these numbers.

Then we need that

`lim_{x\rightarrow(-2)/(3)}f(x)=f((-2)/(3))=k`

As we are approaching -2/3 then we are taking values greater and less than -2/3. Therefore for the limit we use
`f(x)=9x^2-(4)/(3)x-2`.

But observe that
`9((-2)/(3))^2-(4)/(3)(-2)/(3)-2=(26)/(9)`.

Then k must be
`(26)/(9)`.

let's verify that in effect f is continuous with
`(26)/(9)`.

`lim_{x\rightarrow (-2)/(3)}f(x)=lim_{x\rightarrow (-2)/(3)} (9x^2-(4)/(3)x-2)=9((-2)/(3))^2-(4)/(3)*(-2)/(3)-2=(26)/(9)=k=f((-2)/(3))`