Answer
$${\text{100% }}$$
Work Step by Step
$$\eqalign{
& {\text{Efficiency }}\left( \% \right) = 100\left[ {1 - \frac{1}{{{{\left( {{v_1}/{v_2}} \right)}^c}}}} \right] \cr
& {\text{Where }}\frac{{{v_1}}}{{{v_2}}}{\text{ is the ratio of the uncompressed gas, and }}c > 0 \cr
& {\text{Let }}r = \frac{{{v_1}}}{{{v_2}}} \cr
& {\text{Efficiency }}\left( \% \right) = 100\left[ {1 - \frac{1}{{{r^c}}}} \right] \cr
& {\text{The limit of the efficiency as the compression ratio }} \cr
& {\text{approaches infnity is:}} \cr
& {\text{Efficiency }}\left( \% \right) = \mathop {\lim }\limits_{r \to \infty } 100\left[ {1 - \frac{1}{{{r^c}}}} \right] \cr
& = 100\mathop {\lim }\limits_{r \to \infty } \left[ {1 - \frac{1}{{{r^c}}}} \right] \cr
& = 100\left( {1 - \frac{1}{{{\infty ^c}}}} \right),{\text{ }}c > 0 \cr
& = 100\left( {1 - \frac{1}{0}} \right) \cr
& = 100 \cr
& {\text{Then the efficiency is 100% }} \cr} $$
