#math/calculus Assume $\lim_{x\to x_{0}}f(x)=L$ and $\lim_{x\to x_{0}}g(x)=M$. Whenever the expressions on the right are defined, $ \begin{aligned} \lim_{x\to x_{0}}(f+g)&=L+M, &\lim_{x\to x_{0}}(cf)&=cL,\\ \lim_{x\to x_{0}}(fg)&=LM, &\lim_{x\to x_{0}}\frac{1}{f}&=\frac{1}{L}\quad(L\neq 0),\\ \lim_{x\to x_{0}}\frac{f}{g}&=\frac{L}{M}\quad(M\neq 0), &\lim_{x\to x_{0}}[f(x)]^{n}&=L^{n}. \end{aligned} $ Also, $\lim_{x\to x_{0}}c=c$ and $\lim_{x\to x_{0}}x=x_{0}$. Repeated use gives the corresponding rules for finite sums and products. The **replacement rule** says that if $f(x)=g(x)$ throughout some punctured interval around $x_{0}$, then $f$ and $g$ have the same limit at $x_{0}$. This justifies canceling a common factor that is nonzero for $x\neq x_{0}$ before taking a limit.