compute the limit as x -------> 9- of:
(x-1)/(x-9) * (int(t^t dt) from x to 9)
(x-1)/(x-9) * (int(t^t dt) from x to 9)
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Note that this limit can be re-written as:
lim (x-->9-) [(x - 1)/(x - 9) * ∫ t^t dt (from t=x to 9)]
= lim (x-->9-) [(x - 1) * ∫ t^t dt (from t=x to 9)]/(x - 9).
Since ∫ t^t dt (from t=x to 9) --> 0 as x --> 9-, this limit takes the form 0/0. Applying L'Hopital's Rule yields:
lim (x-->9-) [(x - 1) * ∫ t^t dt (from t=x to 9)]/(x - 9)
= lim (x-->9-) [∫ t^t dt (from t=x to 9) + (x - 1)x^x]
(Note that d/dx ∫ t^t dt (from t=x to 9) = x^x from the FTC part 2.)
= 0 + (9 - 1)(9^9)
= 8 * 9^9.
I hope this helps!
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Just a note: 8 * 9^9 does simplify to 3,099,363,912.
lim (x-->9-) [(x - 1)/(x - 9) * ∫ t^t dt (from t=x to 9)]
= lim (x-->9-) [(x - 1) * ∫ t^t dt (from t=x to 9)]/(x - 9).
Since ∫ t^t dt (from t=x to 9) --> 0 as x --> 9-, this limit takes the form 0/0. Applying L'Hopital's Rule yields:
lim (x-->9-) [(x - 1) * ∫ t^t dt (from t=x to 9)]/(x - 9)
= lim (x-->9-) [∫ t^t dt (from t=x to 9) + (x - 1)x^x]
(Note that d/dx ∫ t^t dt (from t=x to 9) = x^x from the FTC part 2.)
= 0 + (9 - 1)(9^9)
= 8 * 9^9.
I hope this helps!
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Just a note: 8 * 9^9 does simplify to 3,099,363,912.