tutorial/ApproximateFunctionsAndInterpolation

Polynomials well-behaved in an interval can misbehave outside

LegendreP[12,0,x]

xmax=1;
Plot[LegendreP[12,0,x],{x,-xmax,xmax},PlotRange->All]

Rational function can misbehave over a tiny interval

xtiny=.01;
Plot[((x-xtiny)/x),{x,-1,1},PlotRange->{0,2}]