5. Yucno A Ha3biBaercs f(x) B Toyke X_(0) HaGnerca Takoe rTenbHoe delta Xneq X_(0) "BOPSIIOIIIIX YCIIOBHIO vert overrightarrow (o)-overrightarrow (o)_(0)vert lt varepsilon , BbITIOJIHCHO 1) vert f(x)+Avert gt varepsilon 2) vert f(x)-Avert gt varepsilon 3) vert f(x)-Avert =0 4) vert f(x)-Avert lt varepsilon

The correct answer is 4) $\vert f(x)-A\vert \lt \varepsilon $.<br /><br />In the given problem, we are given that $f(x)$ is a function and $X_{0}$ is a point in the domain of $f(x)$. We are also given that $\delta $ is a positive real number and $X\neq X_{0}$. We are asked to find the condition that satisfies the given inequality $\vert \overrightarrow {o}-\overrightarrow {o}_{0}\vert \lt \varepsilon $.<br /><br />The inequality $\vert \overrightarrow {o}-\overrightarrow {o}_{0}\vert \lt \varepsilon $ represents the distance between the vectors $\overrightarrow {o}$ and $\overrightarrow {o}_{0}$ is less than $\varepsilon $. This means that the point $X$ is within a distance of $\varepsilon $ from the point $X_{0}$.<br /><br />To satisfy this condition, we need to find a value of $A$ such that $\vert f(x)-A\vert \lt \varepsilon $. This means that the distance between the function $f(x)$ and the constant $A$ is less than $\varepsilon $.<br /><br />Therefore, the correct answer is 4) $\vert f(x)-A\vert \lt \varepsilon $.