How do you show that the function #h(x)=xe^sinx# is continuous on its domain and what is the domain?

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Answer 1 · 2016-11-18T02:16:52

# h(x) = xe^sinx #

#sinx# is continuous over #RR# and it's domain is #x in R#, and it's range is #x in [-1,1]#

#e^x# is continuous over #RR# and it's domain is #x in RR#, and it's range is {X in RR | x>0}.

#x# is continuous over #RR# and it's domain is #x in RR#, and it's range is #x in RR#.

Consequently, #e^sinx# is continuous over #RR#, and it's range is #x in RR#, and it's domain is #{x in RR | e^-1<=x<=e}#

Hence , #h(x)=xe^sinx # is continuous over #RR#, and it's range is #x in RR#. and it's domain is #x in RR#.

In fact #h(x)# oscillates between #y=e^-1# and #y=ex#

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