Find the limit multiple choice

Question

Find the limit: lim_x->0[(√x+4)-2]/x

a. 0

b.1/9

c.1/4

d.1/6

Answer 1

Here is a non-algebraic approach:

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

Let f(x) = √(x+4). Then f(0) = 2

So, the given limit can be expressed as lim_x→0 [(f(x) - f(0)) / (x - 0)] = f'(0)

Since f'(x) = 1 / [2√(x + 4)], we have f'(0) = 1/4.

Answer 2

lim_x->0[(√x+4)-2]/x = lim_x->0[√x+4)-2][(√(x+4)+2\/x[(√x+4) +2] = lim_x->01/[√(x+4)+2] = 1/4