Download e-book for kindle: Analysis 2 (Springer-Lehrbuch) by Wolfgang Walter

By Wolfgang Walter

ISBN-10: 3540429530

ISBN-13: 9783540429531

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Extra info for Analysis 2 (Springer-Lehrbuch)

Example text

Suppose it is claimed that limx→0 f (x) = 0. Analyze this proposal; what value for shows it won’t fly? Can you do it numerically? Graphically? 47: Things seem clear, calculator aided. Can we convince a calculator skeptic? Try this: The function seems to return to the value 1 (in fact, infinitely often, even in the interval (0, 1), say). For what x does sin(1/x) = 1? 48: As before, a failed attempt at δ won’t do; we must show that no possible δ works. 5 An x such that sin(1/x) = 1 is a great candidate to show that some proposed δ was unsuccessful.

37: Every function f has a maximum point. 38: . . for every > 0 there exists δ greater than zero . . 39: Every positive number has a square root. , polynomials). Perhaps surprisingly, we won’t prove these theorems. One sees the proofs of these theorems in a first course in real analysis or advanced calculus; there one grapples not only with limits at quite a technical level, but with crucial and somewhat subtle properties of the real number line. If you are like many students, you believe the real numbers consist of reasonable √ things like the rational numbers (fractions), 2 and the like, π, e, and .

Does this function have a limit at 0? If it does, argue graphically why, and try to find δ at least for various numerical ’s, and if possible for a general . If not, argue graphically and using the appropriate form of argument that it does not, and be as explicit as possible about δ’s and values of x. 58: Repeat the previous exercise with another variant of one of the functions above: define f by f (x) = 1 ), x = 0, x2 · sin( x 0, x = 0. 59: Repeat with yet another variant of one of the functions above: f (x) = x, x rational, 0, x irrational.

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Analysis 2 (Springer-Lehrbuch) by Wolfgang Walter

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