Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

Question

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

f(x) = x+ rad (x-2), [2, infinity]

Answer 1

To begin, we need to make sure that the function exists at x=2, which it clearly does: f(2) = 2. Next we need to ensure that 1) the limit at any point above 2 exists, 2) the function exists at those points, and 3) that these first two values are equal. To show this let A be any real value in the interval (2, ∞).

1) Limit exists; there is no domain issue that would cause a vertical asymptote or hole in the graph.

lim_x→A x+√(x-2) = A + √(A-2)

2) The function exists. Same as before, no domain issue causing a lack of existence in the graph.

f(A) = A + √(A-2)

3) Clearly f(A) = lim_x→A f(x).

So this function is continuous on the interval [2, ∞).