Finding the Limit of a Power

Find:

1. \(\lim\limits_{x\rightarrow 2}x^{5}\)
2. \(\lim\limits_{x\rightarrow 1}( 2x-3) ^{3}\)
3. \(\lim\limits_{x\rightarrow c}x^{n}\)

Solution

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