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本题考查极限的求解。$\lim _{n\rightarrow \infty }\dfrac {(n+1)(n+2)(n+3)}{5{n}^{3}}$$=\lim _{n\rightarrow \infty }\dfrac {(1+\frac {1}{n})(1+\frac {2}{n})(1+\frac {3}{n})}{5}$$=\lim _{n\rightarrow \infty }\dfrac {(1+1+\frac {2}{n})(1+\frac {3}{n})}{5}$$=\lim _{n\rightarrow \infty }\dfrac {(2+\frac {2}{n})(1+\frac {3}{n})}{5}$$=\lim _{n\rightarrow \infty }\dfrac {2+2\cdot \frac {1}{n}+1+\frac {3}{n}+\frac {2}{n}+\frac {6}{n^{2}}}{5}$$=\lim _{n\rightarrow \infty }\dfrac {3+3\cdot \frac {1}{n}+\frac {1}{n}+\frac {6}{n^{2}}}{5}$$=\lim _{n\rightarrow \infty }\dfrac {3+4\cdot \frac {1}{n}+\frac {6}{n^{2}}}{5}$$=\dfrac {3}{5}$

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