Identifying Indeterminate Forms of the Types \(\dfrac{0}{0}\) and \(\dfrac{\infty }{\infty }\)

1. \(\dfrac{\cos ( 3x) -1}{2x}\) is an indeterminate form at \(0\) of the type \(\dfrac{0}{0}\) since \[\lim\limits_{x\rightarrow 0}[ \cos ( 3x) -1] =0\, \hbox{and}\, \lim\limits_{x\rightarrow 0}\,(2x) =0\]
2. \(\dfrac{x-1}{x^{2}+2x-3}\) is an indeterminate form at \(1\) of the type \(\dfrac{0}{0}\) since \[ \hbox{\(\lim\limits_{x\rightarrow 1}( x-1) =0\) and \(\lim\limits_{x\rightarrow 1}( x^{2}+2x-3) =0\)} \]
3. \(\dfrac{x^{2}-2}{x-3}\) is not an indeterminate form at \(3\) of the type \(\dfrac{0}{0}\) since \(\lim\limits_{x\rightarrow 3}( x^{2}-2) \neq 0\).
4. \(\dfrac{x^{2}}{e^{x}}\) is an indeterminate form at \(\infty \) of the type \(\dfrac{\infty }{\infty }\) since \[ \hbox{\(\lim\limits_{x\rightarrow \infty}x^{2}=\infty \) and \(\lim\limits_{x\rightarrow \infty }e^{x}=\infty \)} \]