### Limit

Lim sqrt (x+3) - sqrt (3)/x x->o

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Lim sqrt (x+3) - sqrt (3)/x x->o

Complex Variables Power Series (I) Abel's Theorem: ∞ If the power series ∑ an zn converg

Lim x-->0 (x + ln(1-x))/(x - ln(1+x))

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(4) (a) Let I1,I2,I3... be open intervals and let J be a closed interval and let J be a closed inteval. Let lk be the length of Ik, and let L be the length of J....Please see the attachement

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Prove rigorously: Let N be an integer > or equal to 2, and let Xsub0....Xsubn E [0,1). Prove that there exist i and j with i not equal to j such that abs (xsubi-xsubj) < 1/n.

Let x be an irrational number. Prove that there exist infinitely many fractions (p/q) with p and q as integers such that: abs(x-[p/q]) < 1/(q^2)

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29.18 Let f be a differentiable on R with a = sup {|f ′(x)|: x in R} < 1. Select s0 in R and define sn = f (sn-1) for n ≥ 1. Thus s1 = f (s0), s2 = f(s1), etc Prove that (sn) is a convergence sequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| ≤ aּ|sn - sn-1| for n ≥ 1.

29.12 (a) Show that x < tan x for all x in (0, π/2). (b) Show that x/ sin x is strictly increasing function on (0, π/2). (c) Show that x ≤ (π/2)ּsin x for all x in [0, π/2].

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Let f be defined on R and suppose it satisfies |f(x + y) - f(x)| ≤ |y|^(3/2) for all x, yER. Show that f is a constant function.

Let ... a) Show that f can be represented by a power series. What is its interval of convergence? b) Calculate the power series expansion for the function F(x) = ... Please see attached for equations.

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Determine the limits of the following sequences if they exist. Justify your answers. (See attachment for full question)

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If (a_n)->0 and Absolute value of b_n -b<=a_n then show that (b_n)->b.

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A-a function f:(a,b)->R is increasing on (a,b) if f(x)<=f(y) whenever x<y in (a,b). Assume f is differentiable on (a,b). Show that f is increasing on (a,b)if and only if f'(x)>=0 for all x belong to (a,b). b-show that the function g(x){x/(2+x^2 sin(1/x)) if x not=0 0 if x=0 is differentiable on R and satisfies g'(0)>0.Now

Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

Prove that if f and g are differentiable functions on an interval A and satisfy f'(x)=g'(x) for all x belong to A, then f(x)=g(x)+k for some constant k belong to R.

Let (f_n) be a sequence of diffrentiable functions defined on the closed interval [a,b] and assume (f'_n) converges uniformly on [a,b]. Prove that if there exists a point xo belong to [a,b] where f_n(xo) is convergent, then (f_n) converges uniformly on [a,b].

G(x)=Sum sign(m top n=0 bottom)(1/2^n)h(2^n x).for more inf. please check #30026,#30028,#30029. show that (g(x_m)-g(0))/(x_m - 0)=m+1, and use this to prove that g'(0) does not exist. any temptation to say something like g'(0)=oo should be resisted. setting x_m=-(1/2^m) in the previous argument produces difference heading to

Taking the continuity of h(x) as given in#30026,#30028 by using any of the functional limits and continuity theorems prove that the finite sum g_m (x)=sum sign(oo top n=0 bottom) of 1/2^n h(2^n x) is continous on R