Precalculus (6th Edition) Blitzer

The limit of a sum, $\left( f\left( x \right)+g\left( x \right) \right)$ is $\underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$.
For finding the limit of a sum, first find the limit of each function in the sum. Then add each of these limits. In other words, the limit of the sum of two functions equals the sum of their limits. In limit notation, If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L$ and $\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=M$, then \begin{align} & \underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to a}{\mathop{\lim }}\,g\left( x \right) \\ & =L+M \end{align} For example: Let $f\left( x \right)=x$ and $g\left( x \right)=2$. To find the limit of the sum of $f\left( x \right)$ and $g\left( x \right)$, $\underset{x\to 2}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)$, \begin{align} & \underset{x\to 2}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)=\underset{x\to 2}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to 2}{\mathop{\lim }}\,g\left( x \right) \\ & =\underset{x\to 2}{\mathop{\lim }}\,x+\underset{x\to 2}{\mathop{\lim }}\,2 \\ & =2+2 \\ & =4 \end{align}