Limit of a function
Exponential functions

lim x → 0 (1 + x) 1 x = lim r → ∞ (1 + 1 r) r = e lim x → 0 e x − 1 x = 1 lim x → 0 e a x − 1 b x = a b lim x → 0 c a x − 1 b x = a b ln c lim x → 0 + x x = 1 {\displaystyle {\begin{array}{lcl}\displaystyle \lim _{x\to 0}(1+x)^{\frac {1}{x}}&=&\displaystyle \lim _{r\to \infty }\left(1+{\frac {1}{r}}\right)^{r}=e\\\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}&=&1\\\displaystyle \lim _{x\to 0}{\frac {e^{ax}-1}{bx}}&=&\displaystyle {\frac {a}{b}}\\\displaystyle \lim _{x\to 0}{\frac {c^{ax}-1}{bx}}&=&\displaystyle {\frac {a}{b}}\ln c\\\displaystyle \lim _{x\to 0^{+}}x^{x}&=&1\end{array}}}
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