4. We say that a function f(x) is bounded if there exists real numbers m and M such that forall real numbers 1, m≤ f(x) ≤ M. (Examples include sin(x) and cos(x), letting m = -1and M = 1.)(a) Let f(z) be any bounded function and g(z) be a function such that lim g(x) = 0. Explainwhy the following argument is incorrect:-blim f(x)g(x)1-b=[lim f(x)] [lim g(x)][lim f(x)].00

Answered: Image /qna-images/answer/aee1801e-ff2f-46af-a510-8217dd2d3ce2.jpg

Answered: 4. We say that a function f(z) is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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21. Discuss the continuity of : 1 x + for x 2
1. Study the continuity of f (x) at x = 2 in each of the following; 3x + 1 if x 2 x3 – 1 if x 2 if x = 2 (x+5 if x > 2 x if x<2 c) f(x) =-
Let f be the function whose graph is given below. (a). Use this graph to find the largest number &> 0 such that if 0 0 such that if 0 < x-4 <8 (Enter dne if no such & exists.) 8= dne then f(z)-4) <3. then f(x) - 4 <0.8.
you need a chart first but do not write it in the solution please.
1. Let a, b be real numbers for which b > a. (a) If f(x) is a function that is continuous at every point on its domain of [a, b), must f attain both a maximum and a minimum value? Explain. (Hint: The fact x = b is excluded is important.) (b) Simple yes or no (ie. explanation not required): if p(x) is a function that is continuous at every point in its domain of [a, b], must p attain both a maximum value and a minimum value? (c) If q(x) is a function that is discontinuous at some point(s) in its domain of [a, b], can q lack either a maximum value or a minimum value? Explain. (d) If q(x) is as in part(c), must q lack either a maximum value or a minimum value? Explain.
8. Suppose that lim, o(f(x) – g(x)) = 0 and h(x) is a function with domain R. a) Assume that |h(x) – h(y)| < 3|x – y| for any x, y E R, if lim,o(h(f (x)) – h(g(x))) = 0 is always true, provide a proof; if not, provide a counterexample. b) Assume that h(x) is continuous for any r E R, if lim,-o(h(f (x)) – h(g(x))) = 0 is always true, provide a proof; if not, provide a counterexample.
For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→ Z a(x) = x. B: Z {0, 1,-1} B(n) = -1 (c) (Here Ro is the set of non-negative real numbers. ) | − | : R → R≥o - >0 |x| if n is even, if n is positive, otherwise. X -X 7: R→R y(x) = X if x ≥ 0, if x ≤ 0.
Suppose f: (0, ∞) → (0, ∞) is a continuous function, and g: (0, ∞) → (0,00) is defined by the property that g(x) = (f(x))² for every positive real number . True or false: If g is differentiable, then f must be differentiable too. True False

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