Suppose f(x) < g(x) for all x E (c – p, c+ p), except possibly at c itself. (i) Prove that lim,→c f(x) < limz+c9(x), provided each of these limits exist. (Hint: prove by contradiction. Suppose lim, c f (x) = L > M = lim,→c 9(x). Let e = and try to obtain a contradiction using the 8-e definition for limits. ) L-M 2 Text (ii) Suppose f(x) < g(x) for all æ E (c – p, c+ p), except possibly at c itself. Does it follow that lim,c f(x) < lim,→c9(x)?

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A: Our objective is to find the limit algebraically.

Q: (4) Let f(x) be the function whose domain is all of R given by the rule 42 if r € Z and f(x) = 10…

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Q: If f(x+ y)= f(x)f (y) for all x, y eR and f (x)=1+g(x)G(x), where lim g (x)=0 and lim G (x)…

A: The problem has been solved in detail in the next step.

Q: 6. Consider a function f with the following property: if g is any function for which lim g(x) does…

A: One side is not true i have given the contradiction.

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A: Please see the below picture for detailed solution.

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A: Above question is solved.

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Q: suppose f,g an d h are functions which g(x)<=f(x) <=h(x) ,for all x in an open interval containing a…

A: No. For the given continuous open interval domain, the limit of f(x) does not exist.

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