Final answer:
The linear approximation of the function does not exist at x = 0.
Step-by-step explanation:
To find the linear approximation of the function g(x) = 5 + 1/x at x = a = 0, we can use the formula for linear approximation: l(x) = f(a) + f'(a)(x - a). First, we need to find f(a) and f'(a). For f(a), we substitute a = 0 into g(x) to get f(a) = 5 + 1/0, which is undefined. However, we can use a limit to find f'(a). Taking the derivative of g(x), we get g'(x) = -1/x^2. Taking the limit as x approaches 0, we have f'(a) = lim(x->0)(-1/x^2) = -1/0^2 = -1/0, which is also undefined. Therefore, the linear approximation l(x) does not exist for this function at a = 0.