Laplace transforms for which value of s?

Haku
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Homework Statement
In each case, state the values of s for which the
Laplace transform exists.
Relevant Equations
Laplace transform
I was wondering how you work out what values of s a Laplace transform exists? And what it actually means? The example given in class is an easy one and asks to calculate the Laplace transform of 3, = 3 * Laplace transform of 1 = 3 * 1/s. Showing this via the definition, where does the range of s come out of? I.e. how can I define when the Laplace transform of 3 = 3/s?
And what happens outside that range?
Thanks.
 
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I do not read the original question so as a general observation, s>0 because factor ##e^{-st}## in Laplace integral should diminish for large t>0.
 

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