The sum of two functions

Suppose now that f and g have a common domain and

$$\begin{array} {rcl} \lim_{x\rightarrow a}f(x) & = & F\\ \lim_{x\rightarrow a}g(x) & = & G\end{array}$$

We want to verify that

$$\lim_{x\rightarrow a}(f+g)(x) = F+G$$

We begin as before, realizing that we can control |f(x) - F| and |g(x) - G|:

$$\begin{array} {rcl} \vert(f+g)(x) - (F+G)\vert & = & \vert f... ...\ & \leq & \vert f(x) - F\vert + \vert g(x) - G\vert\end{array}$$

Now here is a plan: There is some m_f so that if 0 < |x-a| < m_f then |f(x) - F| < t/2 and there is some m_g so that if 0 < |x-a| < m_g then |g(x) - G| < t/2. So if $0 < \vert x-a\vert < \min(m_f,m_g)$ then |(f+g)(x) - (F+G)| < (t/2) + (t/2) = t, as required.