without using L hop-ital rule and series expansion , $ \lim_{t\rightarrow 0}\left(t\cot t+t\ln t\right)$ without using L hop-ital rule and series expansion , $\displaystyle \lim_{t\rightarrow 0}\bigg(t\cot t+t\ln t\bigg).$
$\displaystyle \lim_{t\rightarrow 0}\bigg(t\frac{\cos t}{\sin t}+\ln t^t\bigg) = 1+\lim_{t\rightarrow 0}\ln(t)^t$
could some help me with this , thanks

Note that $t \to 0^+$ 
$$x\to 0 \to \ln(1+x)\sim x$$
$$\displaystyle \lim_{t\rightarrow 0^+}\bigg(t\frac{\cos t}{\sin t}+\ln t^t\bigg) =\\ 1+\lim_{t\rightarrow 0^+}t\ln(t)=\\
1+\lim_{t\rightarrow 0^+}t\ln(1+(t-1))=\\
1+\lim_{t\rightarrow 0^+}t\times (t-1)=\\1+0$$
