How to verify limn→∞∫∞0e−nxsin(ex)dx=0?
My idea is to use the dominant convergence theorem with fn(x):=e−nxsin(ex) and f(x):=limn→∞fn(x).
⇒limn→∞∫∞0e−nxsin(ex)dx=∫∞0limn→∞e−nxsin(ex)dx=∫∞0limn→∞0dx=0
Can I use this here?
Answer
How to verify limn→∞∫∞0e−nxsin(ex)dx=0
You may just observe that
|∫∞0e−nxsin(ex)dx|≤∫∞0|e−nx|dx=1n,(n>0), and then let n→+∞.
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