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Sheet (6) solution

# Sheet (6) solution - Math 103 Class Work Sheet(6...

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1 Math 103 Class Work Sheet (6) Solution [1] (a) 0 0 1 1 lim 2 9 1 x x x so we apply L’Hopital rule 2 9 2 9 lim 1 1 lim 8 1 2 9 1 x x x x x x (b) 0 0 sin 1 cos lim 2 x x x so we apply L’Hopital rule  x x x x x x x x tan lim cos sin lim sin 1 cos lim 2 2 2 (c) 0 0 1 lim 3 0 t e t t so we apply L’Hopital rule 3 1 3 lim 1 lim 3 0 3 0 t t t t e t e (d) x e x x lim so we apply L’Hopital rule 1 lim lim x x x x e x e (e) x x x ) ln(ln lim so we apply L’Hopital rule (f) 0 0 1 lim 2 0 x x e x x so we apply L’Hopital rule 0 0 2 1 lim 1 lim 0 2 0 x e x x e x x x x so we apply L’Hopital rule 2 1 2 lim 2 1 lim 0 0 x x x x e x e 0 1 ln 1 lim 1 1 . ln 1 lim ) ln(ln lim x x x x x x x x x (g) 0 0 cos 1 lim 2 0 x x x so we apply L’Hopital rule 0 0 2 sin lim cos 1 lim 0 2 0 x x x x x x 2 1 2 cos lim 2 sin lim 0 0 x x x x x (h) 0 0 ) 4 ( tan lim 1 0 x x x so we apply L’Hopital rule 4 1 4 . ) 4 ( 1 1 1 lim ) 4 ( tan lim 2 0 1 0 x x x x x (i) ) .( 0 ) ln( . lim 0  x x x write the product as quotient x x x x x x 1 ln lim ) ln( . lim 0 0 so we apply L’Hopital rule 0 2 lim 2 1 1 lim 1 ln lim 2 1 0 2 3 0 0 x x x x x x x x (j) ) .( 0 ln . sin lim 0  x x x write the product as quotient x x x x x x csc ln lim ln . sin lim 0 0 so we apply L’Hopital rule 0 0 cos sin lim cot csc 1 lim csc ln lim 2 0 0 0 x x x x x x x x x x x so we apply L’Hopital rule 0 1 0 0 cos sin cos sin 2 lim cos sin lim 0 2 0 x x x x x x x x x x

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2 (k) 0 . 1 tan lim x x x write the product as quotient 0 0 1 1 tan lim 1 tan lim x x x x x x so we apply L’Hopital rule (l) x x x csc 1 lim 0 write the difference as quotient 0 0 sin sin lim csc 1 lim 0 0 x x x x x x x x so we apply L’Hopital rule 0 0 sin cos 1 cos lim sin sin lim 0 0 x x x x x x x x x x so we apply L’Hopital rule 0 cos cos sin sin lim sin cos 1 cos lim 0 0 x x x x x x x x x x x 1 1 sec lim 1 1 . 1 sec lim 1 1 tan lim 2 2 2 2     x x x x x x x x x (m) x x x x 2 lim write the difference as quotient or 0 0 1 1 1 1 lim 1 1 1 lim lim lim
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Sheet (6) solution - Math 103 Class Work Sheet(6...

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