Why can't I get the MGF for exponential function? What am I doing wrong?

Assume hazard rate = 1

=∫ exp(t*x)*exp(-x) dx

limit x-> infinity { [exp(t*x-x)]/(t-1) }

which does not = 1/(1-t)

Please help

Assume hazard rate = 1

=∫ exp(t*x)*exp(-x) dx

limit x-> infinity { [exp(t*x-x)]/(t-1) }

which does not = 1/(1-t)

Please help

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