Can we cancel the equality mark here?

The function $$g(x)=\int_1^x\frac{1}{t^2+1}{\rm d}t-\int_1^x \frac{1}{t^2+f^2(t)}{\rm d}t$$ Is strictly increasing and $g(1)=0<g(2)<g(x)$ for $x>2$ hence $\lim_{x\to\infty}g(x)\geq g(2)>0$ so$$\lim_{x\to\infty}g(x)=\lim_{x\to\infty}(\frac\pi4-(f(x)-1))=\lim_{x\to\infty}(\frac\pi4+1-f(x))>0$$

So $$\lim_{x\to\infty}f(x)<\frac\pi4+1$$