MthT 430 Chapter 5a Projects - Limits

430template.pdf

Limits

1.

Let F(x) be a function such that

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domain(F) = R.

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For all x,y, F(x + y) = F(x) ·F(y).

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F(0) ≠ 0.

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lim x → 0  F(x) − F(0) x = π.

Find

lim x → 0  F(a + x) − F(a) x .

2.

Let G(x) be a function such that

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domain(G) = R⁺ ≡ {x | x > 0}.

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For all x,y > 0, G(x ·y) = G(x) + G(y).

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G(1) = 0.

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lim x → 0  G(1 + x) x = π2.

For a > 0, find

lim x → 0  G(a + x) − G(a) x .

3.

Let f(x) be a function such that

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domain(f) = [0,1).

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For all x (in [0,1)), 0 ≤ f(x) < 1.

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The function f is increasing on [0,1).

Show that there is a number L, 0 ≤ L ≤ 1, such that

lim x → 1−  f(x) = L.

Hint: Construct a binary expansion for L.

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On 18 Sep 2014, 21:10.