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One way to understand limits is compound interest. Start with a dollar, compound 100% in year to get $2. Halving the interest rate and compounding twice gives $
by quarterwave 12y ago
One way to understand limits is compound interest. Start with a dollar, compound 100% in year to get $2. Halving the interest rate and compounding twice gives $2.25. The "limit" of this less-interest-more-often process, ad infinitum, is e (2.718...). This introduces both infinitesimal and infinite.
- ivan_ah 12y agoExcellent example: e = lim_{n -> ∞} (1 + 1/n)^n The concept of infinity (∞) is perhaps the most important new idea in calculus. Specifically, calculus is about procedures with infinite number of steps, or infinitely small steps. The derivative is a slope calculation (rise/run) with an infinitely short run. The integral is a rectangles-approximation-to-an-area using infinitely thin rectangles, and series are summation procedures with infinite number of steps. High school math deals with procedures with finite number of steps, whereas in calculus we learn to use infinity as part of our calculations. The reason why limits are important is because they allow us to make certain statements that would otherwise not be true: 1/n ≠ 0 even if n is huge but lim_{n -> ∞} 1/n = 0 sum([1/2^n for n in range(0,N)]) = 1.999999... ≠ 2 but sum([1/2^n for n in range(0,∞)]) = 2 The equality in both of the above examples depends on (mentally) carrying out a procedure with infinite number of steps. (examples taken from my math book)