Tangent line, small o

Question

Let $f,g,h \in C^2[a,b], ||f|| = |f(x_0)|$ and let $$ f'(x_0) = 0, ~~ f(x_0)f''(x_0)<0, ~~ g(x_0) = 0, ~~ g'(x_0)\neq 0.$$ I have to find local max $$ ||\phi_\lambda|| = ||f +\lambda g + \frac{\lambda^2}{2}h||_{[x_0-\epsilon, x_0+\epsilon]}$$ I have a proof but I don't understand one thing. Let $$\psi(x,\lambda) = \phi'_\lambda(x) = f'(x) + \lambda g'(x) + \frac{\lambda^2}{2}h'(x) = 0. $$ I know from implicit function theorem that exists a function $x(\lambda)$ such that $$ \psi(x,\lambda) = 0 \Leftrightarrow x_\lambda = x_0 - \frac{g'(x_0)}{f''(x_0)}\lambda + o(\lambda)$$ I understand that I have a pattern on $x_\lambda$ from the equation of the tangent line because $ f''(x_0)(x-x_0)+g'(x_0)(\lambda - 0) =0 $, I know the definition of small o but why we add here $o(\lambda)$? Is this tangent line in neighborhood of $0$?