Limits describe approach. Derivatives describe instantaneous rate. Both start from the same idea: look at nearby values, then shrink the neighborhood.
We write
when can be forced arbitrarily close to by taking close enough to (but not necessarily equal to ). The value itself may be missing or different — the limit only cares about the approach.
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Lesson close · Basic Calculus
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Sequence
Continue to Chain Rule & Nested Logs, the next prerequisite step in this course.
One-sided limits matter when left and right disagree:
The two-sided limit exists only when those match. Infinite limits (vertical asymptotes) say the values grow without bound as approaches ; that is still useful language even though is not a real number.
is continuous at when all three hold:
Break any one and you get a jump, hole, or asymptote. Quants meet the same checklist when a payoff or payoff-like map is discontinuous at a barrier — pricing jumps with the function, not with a hand-waved “average.”
The derivative at is
when that limit exists. The numerator is a change in output; the denominator is a change in input. Shrink and you get instantaneous rate.
Worked check: at .
So . Same answer from the power rule , but the limit is the definition the rule is summarizing.
Interactive · difference quotient → derivative
f(x)=x² at x₀=2 · [f(x₀+h)−f(x₀)]/h = 4.5000 · true f′=4
Thompson's opening move, in Chapter I of Calculus Made Easy (1914), is not an epsilon-delta argument. He defines the two symbols that scare students: means "a little bit of," so is a little bit of , and the integral sign is a long S, "the sum of" those bits. The whole of is the sum of its little bits.
Chapter II ranks smallness. A minute is small beside a day, and a second is small beside a minute. A product of two little bits is small beside either bit alone, so in a relative growth it is dropped. If grows by when grows by , the ratio is what remains after those second-order bits are discarded. That ratio is the derivative.
His falling-body example is feet. A little extra time adds
Drop the square of the little bit. Then feet per second. At , the speed is feet per second. The limit above is the same discard, stated so that "little" has a precise meaning.
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