I have to use the linear approximation of $f(x, y, z) = $($5x^2$ + $y^2$)$/(z + 1)$ at $(-2, 1, 1)$ to estimate $f(-1.98, 0.97, 1.03)$
I'm getting $10.1125$, however it is a bit too low (I guess) if i compare this result with the normal substitution: ~$10,119$
Is it really supposed to be $10.1125$ or I'm doing something wrong?
You have it right. Linearization doesn't give the exact value, it gives an approximation.
For example, if you take the parabola $y = x^2$ at $x = 2$ the slope of the tangent line is $4$, and the linear estimate of $2.03^2$ is therefore $4 + 4*.03 = 4.12$, while the exact answer is 4.1209.