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EXAMPLE 7Finding the Limit of a Power

Find:

  1. (a) lim
  2. (b) \lim\limits_{x\rightarrow 1}( 2x-3) ^{3}
  3. (c) \lim\limits_{x\rightarrow c}x^{n}

Solution

  1. (a) \lim\limits_{x\rightarrow 2} x^{5}=\left( \lim\limits_{\kern.5ptx\rightarrow 2}x\right) ^{5}=2^{5}=32
  2. (b) \lim\limits_{x\rightarrow 1}( 2x-3) ^{3}= \left[ \lim\limits_{\kern.5ptx\rightarrow 1}( 2x-3) \right] ^{3}= \left[ \lim\limits_{\kern.5ptx\rightarrow 1}( 2x) -\lim\limits_{x\rightarrow 1}3 \right] ^{3}\ =( 2 -3) ^{3}=-1
  3. (c) \lim\limits_{x\rightarrow c}x^{n}=\left[ \lim\limits_{\kern.5ptx\rightarrow c}x\right] ^{n}=c^{n}