find the limit .

Question

My try

But after that I stuck . Can anbody help me .

Please explain me as I am a high school student

Answer 1

Note that we can write

$$\begin{align} (1+3t\pm 2t^2)^{1/t}&=e^{\frac1t \log(1+3t\pm 2t^2)}\\\\ &=e^{3\pm 2t -\frac92 t+O(t^2)}\\\\ &=e^3\left(1+\pm 2t-\frac92 t+O(t^2)\right) \end{align}$$

Thus, the limit of interest is

$$\lim_{t\to 0}\left(\frac{(1+3t+ 2t^2)^{1/t}-(1+3t- 2t^2)^{1/t}}{t}\right)=e^3\lim_{t\to0}(4+O(t))=4e^3$$