Evaluate the following limit using Taylor series: $\underset{z\to 0}{\lim }\frac{\ln (1+z)}{z}$

Evaluate the following limit using Taylor series: $\underset{t\to 0}{\lim }\frac{e^t-1-t}{t^2}$

Evaluate the following limit using Taylor series: $\underset{y\to 0}{\lim }\frac{\sin \left(y\right)-\tan \left(y\right)}{y^3}$

Evaluate the following limit using Taylor series: $\underset{y\to 2}{\lim }\frac{y^2-4}{\ln (y-1)}$

Find the third-order Taylor polynomial for the function $f\left(x\right)=\cos \left(3x\right)$ centered at $a=0$.