Question

How do you evaluate the limit `(sqrt(x+4)-2)/x` as x approaches `0`?

Answer 1

`lim_(x->0)(sqrt(x+4)-2)/x =1/4`

Explanation:

Multiply the numerator and the denominator by `sqrt(x+4)+2`

`lim_(x->0)(sqrt(x+4)-2)/x = lim_(x->0)(sqrt(x+4)-2)/x*(sqrt(x+4)+2)/(sqrt(x+4)+2)`

Since `(a-b)(a+b) = a^2-b^2` we have

`lim_(x->0)(x+4-4)/(x(sqrt(x+4)+2)) = lim_(x->0)x/(x(sqrt(x+4)+2))=lim_(x->0)1/(sqrt(x+4)+2) = 1/(sqrt(0+4)+2) = 1/(2+2) = 1/4`