Elimination with Amines

$\hspace{35 mm}$[![Reaction](https://i.stack.imgur.com/jCe9s.png)](https://i.stack.imgur.com/jCe9s.png)


ATTEMPT:
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Since $\ce{Et3N}$ is a weak base and the carbon to which $\ce{Br}$ is attached is tertiary, we can rule out $\mathrm{E2}$ and $\mathrm{S\_N2}.$ As we are providing heat, the reaction is likely to go via $\mathrm{E1}$ forming a stable tertiary carbocation.


Now as the base is bulky it will knock out the most easily available $\ce{H}$, giving us the less substituted alkene.


However my text suggests that this reaction proceed via $\mathrm{E2}$ with the base taking a $\ce{H+}$ and knocking off $\mathrm{Br}$ to give the same final product.


In general with amines out of $\mathrm{E1}$ and $\mathrm{E2}$ which reaction is preferred if there is a possibility of stable carbocation formation?


