Question

When exactly does error tend to zero in calculus?

Answer 1

AA and BB do not literally become overlapped, and the line CC is not obtained literally as a line joining points AA and BB on the curve. The diagram is designed to motivate the plausibility of the fact that as BB gets "sufficiently close" (but not equal) to AA, the line connecting AA and BB gets "arbitrarily close" to the tangent line CC to the curve at the point AA. This is consistent with the idea that ΔxΔx "approaches 00", but is never equal to 00.

More precisely, the slope of the line CC is the derivative of ff at x0x0, f′(x0)f′(x0), defined by

f′(x0)=limΔx→0f(x0+Δx)−f(x0)Δx.
f′(x0)=limΔx→0f(x0+Δx)−f(x0)Δx.
More informally again, this limit means that you can get "arbitrarily close" to f′(x0)f′(x0) by taking ΔxΔx "sufficiently small" (but not 00) in the difference quotient f(x0+Δx)−f(x0)Δxf(x0+Δx)−f(x0)Δx, which is the slope of the line connecting AA to BB in the diagram.

Answer 2

Such a division can be formally expressed as a0 where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives a (assuming a ≠ 0), and so division by zero is undefined.