Trying to find the point wise limit

Question

Here is the function $$f_k(x)=\frac{x^k+3}{x^k+5} ,x\in[0,1]$$ $$f_k(1)=\frac{2}{3}, f_k(0)=\frac{3}{5}$$ Bound by $\frac{2}{3}$

I am confused on how to find it One theorem || fk-f||[0,1]= sup( fk(x):00 Not uniform

Answer 1

I am assuming you are asking about $\lim_{k\to \infty}f_k(x)$
Since $f(1)=\frac{2}{3}$ for all $k$, $\lim_{k\to \infty}f_k(1)=\frac{2}{3}$.
For $x\lt 1$, $x^k\to 0$ (not uniformly), so that $\lim_{k\to \infty}f_k(x)=\frac{3}{5}$.

As you see it is necessary to treat the problem as two separate cases.