format compact
format short
% The first plot
p = [256 0 -576 0 432 0 -120 0 9 -.1]
x = linspace(0,2*pi/3);
f = cos(9*x);
f1 = polyval(p,cos(x));
figure(1);
subplot (1,2,1);
plot(x,f,x,f1)
axis equal
% The first limit!
format rat
x1 = 0.25.^[0:8]
y = (tan(4*x1)-sin(4*x1))./x1.^3;
[x1;y]'
% Clearly, the limit is 32.
% The second limit!
format short
y1 = (1+9*x1).^(1./x1);
y2 = (1+x1).^(9./x1);
[x1; log(y1); log(y2)]'
% The limit is e^9
% Another plot!
f3 = cos(4*x); 
x3 = pi/12; y3 = cos(4*x3); 
m = -4*sin(4*x3);
tangentline = m*(x-x3)+y3;
subplot(1,2,2);
plot(x,f3,x,tangentline,'k',x3,y3,'o');
axis ([0 pi/2 -1 1])