\documentstyle[lbright,example,twocolumn,palatino,a4,11pt]{article}

\begin{document}
\title{Mathematical exercises --- answers}
\author{S Rahtz}
\date{August 1992}
\maketitle

\begin{example}
\begin{equation}
(fg)'=f'g+fg'
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\alpha\beta=\gamma+\delta 
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\Gamma(n)=(n-1)
\end{equation}
\end{example}


\begin{example}
\begin{equation}
X\wedge(Y\vee Z)=
(X\wedge Y)\vee(X\wedge Z)
\end{equation}
\end{example}


\begin{example}
\begin{equation}
2+4+6+...+2n=n(n+1)
\end{equation}
\end{example}

\begin{example}
\begin{equation}
\vec{x}\cdot\vec{y}\not=0 
\mbox{ if and only if } 
\vec{x}\not\perp
\vec{y}
\end{equation}
\end{example}


\begin{example}
\begin{equation}
(\forall x  \epsilon\Re)
(\exists y\epsilon\Re)y >x
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\frac{a+b}{c}\quad  
\frac{a}{b+c} \quad
\frac{1}{a+b+c}\not=
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}
\end{equation}
\end{example}

\begin{example}
\begin{equation}
e^x\quad e^{-x}\quad 
e^{i\pi}+1=0\quad
x_0 \quad x^2_o \quad 
x^{\;2}_o \quad 
2^{x^x}
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\nabla^2 f(x,y)=
\frac{\partial^2 f}{\partial x^2}+
\frac{\partial^2 f}{\partial y^2}
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\lim_{x\rightarrow 0}
(1+x)^{\frac{1}{x}}=e
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\lim_{x\rightarrow 0 }+x^x=1 
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\sqrt{2}\quad 
\sqrt{\frac{x+y}{x-y}}
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\sqrt[3]{10}\quad \sqrt[x]{e} 
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\parallel x\parallel=\sqrt{x} \cdot x 
\end{equation}
\end{example}


\begin{example}
\begin{equation}
c=1+\sqrt{a}^5+{b}^2
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\sqrt[5]{x_1+2x_2-5x_3} 
\end{equation}
\end{example}


\begin{example}
\begin{equation}
x^{\frac{\pi}{\omega+\theta}}
\end{equation}
\end{example}


\begin{example}
\begin{equation}
\lim_{x \to \infty}
\frac{\tan x}{\alpha} = 7 
\end{equation}
\end{example}

\begin{example}
\begin{equation}
\tan (2\theta) = 
\frac{ 2\tan\theta}{ 1-\tan^2\theta.}
\end{equation}
\end{example}



\end{document}

