If we know exists and we want to show f is actually uniformly continuous, it helps to invoke a nice theorem.

Theorem: If is a continuous real function on a closed and bounded set of real numbers, then is in fact uniformly continuous.

Then if we can define a value of at which maintains continuity, we've got it!

.Spoiler:

Conversely if the function is uniformly continuous then a straightforward use of the definition shows the limit exists.

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