What's simplest, somewhat realistic function f(x) where f(x) --> inf as x-->x_lim, where x_lim is finite? No linear system can do this btw
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but the model breaks down before it hits that limit of d-->0
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Slope of an analog clock's hour hand (wrt clock face) goes to (minus) infinity as hour approaches 6 and 12.
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that seems like a cheating since you can fix it by simply rotating the clock. Nothing fundamentally going to infinity.
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reciprocal: f(x) = 1/x, x_lim = 0. Makes sense to consider division primitive since it's axiomatic to a field.
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see other responses... can't think of basic physical process that approximates 1/x well
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