I can't think of a generic way to constrain a function to tend to infinity as x approaches 1, but you might have some luck defining a function that has that behavior and fitting it. For example:
load census % define some data having your shape
x = (cdate-1775)/10;
y = 1./pop;
ft = fittype('b1/((x-1)^b2)'); % has the desired asymptotic behavior
f = fit(x,y,ft,'start',[1 1])
plot(x,y,'bx', x,f(x),'r-') % not a great fit, but it's a start
