지수함수 극한 보기

(ㄱ) $f ( x ^{2} ) = x ^{2}$이면 $\displaystyle \lim\limits _{x \rightarrow 0} \frac{e ^{x} {}^{2} - 1}{x ^{2}} \times x = 0$(참) (ㄴ) $\displaystyle \lim\limits _{x \rightarrow 0} \frac{e ^{x} - 1}{f ( x )} = \lim\limits _{x \rightarrow 0} \frac{e ^{x} - 1}{x} \times \frac{x}{f ( x )} = 1$이므로 $\displaystyle \lim\limits _{x \rightarrow 0} \frac{x}{f ( x )} = 1$ $\displaystyle \lim\limits _{x \rightarrow 0} \frac{3 ^{x} - 1}{f ( x )} = \lim\limits _{x \rightarrow 0} \frac{3 ^{x} - 1}{x} \times \frac{x}{f ( x )} = \ln 3$(참) (ㄷ) 반례) $f ( x ) = | x |$라 하면 $\displaystyle \lim\limits _{x \rightarrow + 0} \frac{e ^{|x|} - 1}{x} = \lim\limits _{x \rightarrow + 0} \frac{e ^{|x|} - 1}{| x |} \times \frac{| x |}{x} = 1$ $\displaystyle \lim\limits _{x \rightarrow - 0} \frac{e ^{|x|} - 1}{x} = \lim\limits _{x \rightarrow - 0} \frac{e ^{|x|} - 1}{| x |} \times \frac{| x |}{x} = - 1$ ∴ $\displaystyle \lim\limits _{x \rightarrow 0} \frac{e ^{|x|} - 1}{x}$은 존재하지 않는다. (거짓)