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![{\displaystyle {\begin{aligned}\lim _{m\rightarrow +\infty }P(1+{\frac {r}{m}})^{mt}&=\lim _{m\rightarrow +\infty }P{\frac {{[(m+r)^{mt}]}'}{{(m^{mt})}'}}\\&=\lim _{m\rightarrow +\infty }P{\dfrac {{\dfrac {\operatorname {d} [(m+r)^{m}]^{t}}{\operatorname {d} [(m+r)^{m}]}}\centerdot {\dfrac {\operatorname {d} [(m+r)^{m}]}{\operatorname {d} (m+r)}}\centerdot {\dfrac {\operatorname {d} (m+r)}{\operatorname {d} m}}}{{\dfrac {\operatorname {d} (m^{m})^{t}}{\operatorname {d} (m^{m})}}\centerdot {\dfrac {\operatorname {d} m^{m}}{\operatorname {d} m}}}}\\&=\lim _{m\rightarrow +\infty }P{\frac {t(m+r)^{(t-1)m}\centerdot (m+r)^{m}[1+\ln {(m+r)}]\centerdot (1+r)}{t[m^{(t-1)m}]\centerdot m^{m}(1+\ln m)}}\\&=\lim _{m\rightarrow +\infty }P(1+r)\\&=P(1+r)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3ea283b8a79ba2ea23a20d997984659cfef535)