(a) Let ƒ : [a, b] → R be continuous with f(a) < 0 and f(b) > 0, and S := {x = [a, b] | f(x) < 0}. If c is the supremum of S, prove that f(c) = 0. (b) Suppose that ƒ : (0,1) → R is uniformly continuous. Prove that limx→o+ f(x) exists.

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Q: 6. Prove that if f RR is continuous with limx→∞ f (x) limx→∞ f (x) f(c) > L for some c E R, then f…

A: The answer provided below has been developed in a clear step by step manner.

Q: calculate the limit of the function limT(æ → 0) [cosx/x2]]

A: To calculate the limit of the function.

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A: Given that a function f(x) and we find the value of limit.

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Q: Solve the partial differential equation xp+yq=z.

A: According to the lagrange linear equation, Pp+Qq=R the solution is,

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A: Let f(x) be the pointwise limit of fn(x) on [a,b].If f∈Ca,b, then f(x)∈Ca,b