Chapter 1.2, Problem 42ES

(a) Explain why the following argument is incorrect.

$\underset{x\to 0}{\lim }\left(\frac{1}{x}-\frac{2}{x^2+2x}\right)=\underset{x\to 0}{\lim }\frac{1}{x}\left(1-\frac{2}{x+2}\right)=\infty \cdot 0=0$

(b) Show that $\underset{x\to 0}{\lim }\left(\frac{1}{x}-\frac{2}{x^2+2x}\right)=\frac{1}{2}$ .