Compute the following limit:

Question

sqrt x+4 -2/x

lim
x-->0

the x+4 is the only thing under the square root.

the answer is .25, but id like to know how to work this out

Answer 1

(sqrt(x+4) -2) /x

L'Hoptials rule:

it is of the form 0/0, so limit equals
1/2sqrt(x+4)= 1/4

Now, the hard way:

multiply numerator and denominator by sqrt(x+4)+2, so limit now is

Lim ((x+4)-4)/x(sqrt(x+4)+2)

= lim 1/sqrt4+2= 1/4