Question

Real analysis

(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case. guction Use 5x2 3x 2 a. lim 3x + 1 (x + 2) x + p sin 2x *Suppose f: (a, b) → R,pE [a, b], and limf(x) > 0 Prove that there exists a δ > 0 such that f(x) > 0 for alls ye(a, b) with 0 < x-pl<.6.2-18-10 진사 10. Suppose E C R,pis a limit point of E, and f:ER. Prove that if f has a limit at p, then there exists a posi tive constant M and a δ > 0, such that lf(x)|s M for all x E E, 0 < pl < δ. 11. a. Prove Theorem 4.1.6(a). b. Prove Theorem 4.1.6(b). 12. *Prove Theorem 4.1.8. Prove Theorem 4.1.9. 13. 14 Let f,g be real-valued functions defined on E C R and let p be a limit point of E. a. If lim fx) and lim (x)()exist, prove that lim gle) exists. b. If limfx) and lim ((a)gGa) exist, does it follow that lim g(r) exists? Let E be a nonempty subset of R and let p be a limit point of E. Suppose f is a bounded real-valued function o E having the property that lim fr) does not exist. Prove that there exist sequences (Pa) and (a) in E with lim p, lim qn p such that lim f(Pa) and lim f(a) exist, but are not equal. *Let f be a real-valued function defined on (a, oo) for some a >0. Define g on (0, ) by 8() f). Prove lim () L if and only if lim &() L 17. Investigate the limits at oo of each of the following functions defined on (0, 0o). 3x2 3x - 1 *a. f(x) =-2x +1 b. f(x)- 2x + 3 d, f(x) = c、f(x)= x+ 1 2x f, f(c) = *e.fo ) = Vtx-x h. f(x) = x sin L where L E R. Prove that linn f(x) = 0.2on 1 Let/: (a, oo)-Abe such that linn xf(x)

10 11 12 13 please

Answer 1

et Fox Then let-O = mịn( , 32). Ten, we have

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129 yng

let- jtr)^.ftx) s hlz), fer all xeE -d 3, we have 2.