Given that \lim_x\to2f(x) = 1,

find \lim_x\to23f(x) + 8x-4.

The first thing we do is break the limit into several pieces:

\lim_x\to23f(x) + 8x-4& = &\lim_x\to23f(x) + \lim_x\to28x-\lim_x\to24.

Now we pull out constants, and evaluate the limits of x and 4:

\lim_x\to23f(x) + \lim_x\to28x-\lim_x\to24& = &3\lim_x\to2f(x) + 8\lim_x\to2x-\lim_x\to24& = &3\lim_x\to2f(x) + 8(2)-4& = &3\lim_x\to2f(x) + 12.

Since \lim_x\to2f(x) = 1,

we find 3\lim_x\to2f(x) + 12 = 3(1) + 12 = 15.