These are the conditions of the continuity in a function:
First, the value of x must have an image.
Second, the lateral limits must be equal:
[tex]\lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)[/tex]
Finally, the value of the limit must be equal to the image of x. This means that:
[tex]f(a)=\lim_{x\to a^}f(x)[/tex]
In this case, we must find a value of k that can make the two lateral limits equal in x =3:
[tex]\lim_{x\to3^+}x^2+k=\lim_{x\to3^-}kx+5[/tex]
We can solve these two limits easily by replacing the x with the value of 3
[tex]3^2+k=3k+5[/tex][tex]\begin{gathered} 9+k=3k+5 \\ 4=2k \\ k=2 \end{gathered}[/tex]
Finally, we can see that the answer is k=2.
