Finding the Limit of \({f(x)=(x+1)^{3/2}}\)

Find \(\lim\limits_{x\rightarrow 8}(x+1)^{3/2}\).

Solution Let \(f(x)=x+1\). Near \(8,\) \(x+1>0,\) so \(( x+1) ^{3/2}\) is defined. Then \[ \begin{eqnarray*} \lim\limits_{x\rightarrow 8}[f(x)]^{3/2}&=&\lim\limits_{x\rightarrow 8}(x+1)^{3/2}\underset{\underset{\color{#0066A7}{\hbox{\(\begin{array}{c}{\lim\limits_{x\rightarrow c}{[f(x)]}^{m/n}{=} \left[ \lim\limits_{\kern.5ptx\rightarrow c}{f(x)}\right]^{m/n}}\end{array}\)}}}{\color{#0066A7}{\uparrow}}}{=}\left[ \lim\limits_{\kern.5ptx\rightarrow 8}(x+1)\right] ^{3/2}=[8+1]^{3/2}=9^{3/2}=27 \end{eqnarray*} \]