Answer
2
Work Step by Step
We know
$$\lim\limits_{\theta \to 0} \dfrac{\mathrm{sin}\left(\theta \right)}{\theta }=1.$$
We will use this fact to compute the limit below.
We want to find
$$\underset{t\to 0}{\lim}\frac{2t}{\mathrm{tan}\left(t\right)}.$$
Note that
$$\frac{2t}{\mathrm{tan}(t)}=\frac{2t}{\left(\frac{\mathrm{sin}\left(t\right)}{\mathrm{cos}\left(t \right)}\right)}=\frac{2t\mathrm{cos}(t)}{\mathrm{sin}\left(t\right)}=\left(\frac{t}{\mathrm{sin}(t)}\right)\left(2\mathrm{cos}\left(t\right)\right).$$
Thus
$$\underset{t\to 0}{\lim}\frac{2t}{\mathrm{tan}(t)}=\underset{t\to 0}{\lim}\left(\left(\frac{t}{\mathrm{sin}(t)}\right)\left(2\mathrm{cos}(t) \right)\right)=\left(1\right)\left(2\right)=2.$$
