# Limits and Absolute Value

Is the following true? Note that a and b are vectors.

$$\lim_{\vec b \to 0}\frac{\vec a \cdot \vec b}{\|\vec b\|} = \lim_{\vec b \to 0}\frac{|\vec a \cdot \vec b|}{\|\vec b\|}$$

Update Sorry, instead of putting a "b", I put an "x" instead.

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No, because in some cases $\vec{a}\cdot\vec{b}$ is negative, whereas $|\vec{a}\cdot\vec{b}|$ is never negative. If $\vec{a}$ and $\vec{b}$ somehow go to $0$ as the vector $\vec{x}$ goes to $0$, then you may get a finite number as the limit. If they are constant and the dot product is not $0$, then that first limit will diverge either to $+\infty$ or to $-\infty$. If you're working in a context where it makes sense to identify those with each other, so that there is only one "infinity", then the two limits could in that sense be equal. But more generally, they're not.
However, if $\vec{a}\cdot\vec{b}$ is positive or $0$, then the two expressions will have equal values if they have any values.
Later edit: Now that you've changed $\vec{x}$ to $\vec{b}$, more can be said. If $\vec{b}$ approaches $\vec{0}$ from the direction toward which $\vec{a}$ points, then the limit will be $\|\vec{a}\|$. If it approaches from the opposite direction, it will be $-\|\vec{a}\|$, and it it approaches orthogonally, it will be $0$. So the limit does not exist.