without using L hop-ital rule and series expansion , $ \lim_{t\rightarrow 0}\left(t\cot t+t\ln t\right)$

Question

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

Answer 1

For $0 < t < 1$

$$ -t\ln t = -2t\ln \sqrt{t}= 2t\int_{\sqrt{t}}^1 \frac{ds}{s} \leqslant 2t\frac{1 - \sqrt{t}}{\sqrt{t}} = 2\sqrt{t}(1 - \sqrt{t})\\ -t\ln t = -2t\ln \sqrt{t}= 2t\int_{\sqrt{t}}^1 \frac{ds}{s} \geqslant 2t(1 - \sqrt{t}) $$

Now use the squeeze theorem to show $t \ln t \to 0$.

Answer 2

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