lim([-1/(x+2)]+1/2)/x as x->0

(-1/(x+2) + 1/2)/x

= ((-2+x+2)/(2(x+2)))/x
= (x/(2(x+2)))/x
= 1/(2(x+2))
= 1/4 as x->0

No tricks here; just a little algebra

top and bottom both go to 0

try l'hopital
derivative of top = 1(1)/(x+2)^2 --->1/4
derivative of bottom = 1
1/4 / 1 = 1/4

To find the limit of the given expression as x approaches 0, we can follow these steps:

Step 1: Simplify the expression.
The given expression can be simplified by combining the fractions:

lim([-1/(x+2)] + 1/2) / x
= [(-1+ (x+2)/2*(x+2))] / x
= [(-1+ (x+2))/2*(x+2)] / x
= [(-1 + x + 2)/(2*(x+2))] / x
= (1 + x)/(2*(x+2)) / x

Step 2: Divide the numerator by x and the denominator by x.
By dividing the numerator and denominator by x, we can simplify the expression further:

= (1/x + x/x)/ (2*(x+2)/x)
= (1/x + 1)/ (2*(x+2)/x)

Step 3: Take the limit as x approaches 0.
Now, we can find the limit by substituting 0 for x in the simplified expression:

lim ((1/0) + 1)/(2*(0+2)/0)
Since 1/0 and 2/0 are undefined, this limit does not exist.

Therefore, the limit of the given expression as x approaches 0 is undefined.