Real Analysis : Limits and Continuity of Piecewise Functions

Determine whether or not each of the following limits exists. Discuss also the continuity of each of the following functions at given point c. Give reasons to your answers.

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Limits and Continuity of Piecewise Functions are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Please help with the following problems regarding limitsandpiecewisefunctions.
1) Let f(x) {o if x is a natural odd number
{1 otherwise
Does f(x) have a limit as x approaches infinity? Explain you answer.
2) Let f(x) {1 if x is a natural odd number
{ 1-1/x

1. A piecewise function is given. Use the function to find the indicated limits, or state that a limit does not exist.
(a) lim is over x gd - f(x), (b) lim is over x gd + f(x), and (c) lim is over xgd f(x)
f(x) = { x^2 - 5 if x < 0 }
{ -2 if x >= 0 } : d = -3
(a) -5 (b) -2 (c) does not exist

Prove : Assume f and g are continous functions defined on interval contaning a, and assume that f and g are differentiable on tis interval with the possible exception of the point a. If f(a)=0 and g(a)=0 then lim f'(x)/g'(x)=L as x->a implies lim f(x)/g(x)=L as x->a.

Taking the continuity of h(x) as given in#30026,#30028
by using any of the functional limitsandcontinuity 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

Are these functions Reimann Integrable? I am just learning this topic, so my description may not be accurate. A function is Reimann Integrable if it's Upper Darboux Sums and Lower Darboux suns are equal.
Or stated another way, if U(f, P) - L(f, P) < e
The two functions are piecewisefunctions.
1) f(x) = { 0 when x =

Limit
Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is +∞ or -∞.
1.
2. and lim f(x) where f(x)=
Decide if the given function is continuous at the specific value of x and why?
1. f(x) =
2. f(x)=
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We have just finished up integration and are done with a first course in analysis, so chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book. Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right,