5) Use the Squeeze Theorem to show lim x→0 x^4 cos ( 1 5x^3 ) =0. (8 points) Squeeze Theorem If f(x) ≤ g(x) ≤h(x) when x is near a (except possibly at a) and lim f(x) = lim h(x) = L x→a x→a then lim g(x) = L x→a
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Step 1: We know that -1 ≤ cos(x) ≤ 1 for all x. Show more…
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A graphing calculator is recommended. Use the Squeeze Theorem to show that lim x→0 x^2 cos(10πx) = 0. Illustrate by graphing the functions f(x) = -x^2, g(x) = x^2 cos(10πx), and h(x) = x^2 on the same screen. Let f(x) = -x^2, g(x) = x^2 cos(10πx), and h(x) = x^2. Then -1 ≤ cos(10πx) ≤ 1 ⇒ -x^2 ≤ x^2 cos(10πx) ≤ x^2. Since lim x→0 f(x) = lim x→0 h(x) = 0, by the Squeeze Theorem we have lim x→0 g(x) = 0.
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