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# Uniform Continuity : Epsilon-delta Proof of Continuity of f(x) = x ^ (1/3)

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Prove (or disprove) the following statement: A function f exists that is uniformly continuous on (a,&#8734;) and for which lim as x-> &#8734; of f(x) = &#8734;.

I know that f(x) = x ^ (1/3) (cube root of x) is uniformly continuous on R, and that it's limit as x approaches infinity is infinite. However, I am having trouble proving this using the definition of uniform continuity: For any given e>0 their exists a d>0 such that |f(x) - f(t)| < e for all x,t satisfying |x - t| < d. Can you please help me with this proof?

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Uniform continuity proofs are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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