Prove the following, using the ε-δ definition of the limit of a function: (a) Let f [0, ∞) → R be defined by f(x) = √. Show that lim f(x)ce [0, ∞). Is f a continuous function?X→C=√c for all(Remark: You may use the fact that 0 ≤ a < b if and only if √a < √b. As a hint onhow to play the ɛ games, look at the proof of Proposition 2.2.6 in the textbook.)(b) Let ƒ : R → R be defined by f(x) := cos(x). Show that lim f(x) = cos(c) for all c € R.Is f a continuous function?X→C(Remark: You may use trigonometric identities here, and the fact that |sin(x)| ≤ |x|,and sin(x)| ≤ 1 for all x € R. See Example 3.2.6 in the textbook for the necessaryalgebra; however, you will need explain all of the steps of the proof to receive credit.)

We prove the details using the e-d definition in step 2. Solution has been typeset in LAtex clearly.

(a) Let f: [0, ∞) → R be defined by f(x) = √x. Show that lim f(x) ce [0, ∞). Is f a continuous function? 24x = √e for all (Remark: You may use the fact that 0 ≤ a < b if and only if √a < √b. As a hint on how to play the & games, look at the proof of Proposition 2.2.6 in the textbook.) (b) Let f: R → R be defined by f(x) := cos(x). Show that lim f(x) = cos(c) for all c € R. Is f a continuous function? X-C (Remark: You may use trigonometric identities here, and the fact that |sin(x)| ≤ |x|, and |sin(x)| ≤ 1 for all x € R. See Example 3.2.6 in the textbook for the necessary algebra; however, you will need explain all of the steps of the proof to receive credit.)

(a) Let f: [0, ∞) → R be defined by f(x) = √x. Show that lim f(x) ce [0, ∞). Is f a continuous function? 24x = √e for all (Remark: You may use the fact that 0 ≤ a < b if and only if √a < √b. As a hint on how to play the & games, look at the proof of Proposition 2.2.6 in the textbook.) (b) Let f: R → R be defined by f(x) := cos(x). Show that lim f(x) = cos(c) for all c € R. Is f a continuous function? X-C (Remark: You may use trigonometric identities here, and the fact that |sin(x)| ≤ |x|, and |sin(x)| ≤ 1 for all x € R. See Example 3.2.6 in the textbook for the necessary algebra; however, you will need explain all of the steps of the proof to receive credit.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 25E

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