Limits and Continuity
A function f(x) has a limit L as x→a if we can make f(x) as close to L as desired.
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Key insight: The limit exists only if the left-hand and right-hand limits are equal.
lim(x→a) f(x) = L
Derivatives
The derivative measures the instantaneous rate of change of a function at any given point.
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f'(x) = lim(h→0) [f(x+h) − f(x)] / h
Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
What is the derivative of sin(x)?
cos(x)
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Pure mathematics is, in its way, the poetry of logical ideas. — Albert Einstein