Let f and g be functions with limits s and t at a point p. Given ε, choose β and γ such that f(x) is within ε/2 of s when x is within β of p, and g(x) is within ε/2 of t when x is within γ of p. Let δ be the smaller of β and γ and verify that f(x)+g(x) is within ε of s+t when x is within δ of p.
As a corollary, the sum of continuous functions is continuous.
If f is as above, and c is a constant, choose δ so that f(x) is within ε/c of s, and c*f is within ε of s. If c is 0 the above doesn't work, but if c is 0 then c×f is everywhere 0, and its limit at p is 0×s, or 0.
If c is complex let n be the norm of c, its distance from the origin in the complex plane. Choose δ so that f(x) is within ε/n of s. Multiply through by c and we are still within ε of c×s.
As a corollary, any linear combination of continuous functions is continuous. Similarly, a linear combination of uniform functions is uniform. The ε δ algebra is the same; it just doesn't depend on p.
Limits can also be added and scaled as x approaches + or - infinity. Instead of restricting x to within δ of p, restrict x to points > m. I'll leave the details to you.