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EXAMPLE 1Identifying Indeterminate Forms of the Types 00 and ∞∞
  1. (a) cos(3x)−12x is an indeterminate form at 0 of the type 00 since lim
  2. (b) \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. (c) \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. (d) \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 \(\lim\limits_{x\rightarrow \infty }e^{x}=\infty \)}