## Precalculus (6th Edition) Blitzer

The limit of a quotient, $\frac{f\left( x \right)}{g\left( x \right)}$ is $\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)}$ such that $\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)\ne 0$
For finding the limit of a quotient, first find the limit of each function in the quotient. Then divide each of these limits, provided that the limit of the denominator is not zero. In other words, the limit of the quotient of two functions equals the quotient of their limits, provided that the limit of the denominator is not zero. In limit notation, If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L$ and $\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=M$, $M\ne 0$ then $\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)}=\frac{L}{M},M\ne 0$ For example: Let $f\left( x \right)=x$ and $g\left( x \right)=2$. To find the limit of the quotient of $f\left( x \right)$ and $g\left( x \right)$, $\underset{x\to 2}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}$, \begin{align} & \underset{x\to 2}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to 2}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to 2}{\mathop{\lim }}\,g\left( x \right)} \\ & =\frac{\underset{x\to 2}{\mathop{\lim }}\,x}{\underset{x\to 2}{\mathop{\lim }}\,2} \\ & =\frac{2}{2} \\ & =1 \end{align}