Evaluate lim x→0 √(x+ 2) − √(2) / x. Do not draw a graph or make a table of values.

Answer 1

Recall that f'(c) = lim_x→c[(f(x) - f(c)) / (x - c)]

So, if we let f(x) = √(x+2) and c = 0, then the given limit is f'(0)

For f(x) = √(x+2), f'(x) = 1/[2√(x+2)]

The value of the limit is f'(0) = 1/[2√2] = √2/4

Answer 2

The first term of the function, √x+2, approaches a finite value, namely √2. Contrarily, the 2nd term, √2/x, increases without bound as x -> 0. Nothing prevents that term from diverging, so the limit does not exist.