What is the Limit of a Rational Function?

[efekwulsemmay]

Homework Statement

All the problem is asking me to do is find the limit. I have to do it algebraicly too which sucks. I can't figure out how to do it.
$$\lim_{x\rightarrow0} \frac{\frac{1}{x+2}-\frac{1}{2}}{x}$$

Homework Equations

I am not sure really.

The Attempt at a Solution

One thing I have tried is to multiply by $$\frac {x}{x}$$. Which gave me:
$$\lim_{x\rightarrow0} \frac{\frac{1}{x+2}-\frac{1}{2}}{x} \times \frac {x}{x}$$

$$=\frac{\frac{1\times x}{x+2}-\frac{1\times x}{2}}{x\times x}$$

$$= \frac{\frac{x}{x+2}-\frac{x}{2}}{x^{2}}$$

$$=\frac{x}{x+2}\rightarrow \frac{x}{x}+\frac{x}{2} \rightarrow 1+\frac{x}{2}$$

$$=\frac{1+\frac{x}{2}-\frac{x}{2}}{x^{2}}$$

$$=\frac{1}{x^{2}}$$

1) I am not sure of my algebra during this and
2) I don't know where to go from here should my algebra check out.

I have already tried to multiply by $$\frac{\sqrt{x}}{\sqrt{x}}$$ but it just seems to give me $$\frac{\sqrt{x}}{x}$$ which doesn't help. I am stuck help me please?

 

[Mark44]

Combine the two fractions in the numerator and then simplify the whole thing.

 

[efekwulsemmay]

Thank you! :):):):):):):):)