Problem Set 12s

0 41 4 sin 4 x 4 cos 4 x lim lim 2 2 x 0 tan 5 x x

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Unformatted text preview: x # & x sin x = − lim % ( x→0 $ x cos x + sin x ' 0 again this is of type , apply L'Hospital's Rule again. 0 f. x2 +1− 6 =5 x →2 x−2 lim ln x sin πx Solution: ln x H 1/ x 1 1 lim = lim = = − x →1 sin πx x →1 π cos (πx ) π (− 1) π ln x 1 lim = − x →1 sin πx π sin 4 x c. lim x → 0 tan 5 x Solution: 0 This limit has the form . 0 4(1) 4 sin 4 x 4 cos 4 x lim = lim = 2 = 2 x →0 tan 5 x x → 0 5 sec (5 x ) 5 51 sin 4 x 4 lim = x →0 tan 5 x 5 1 ex −1− x − x2 2 d. lim 3 x →0 x Solution: 0 This limit has the form . 0 12 ex −1− x − x x x Hx H 2 = e − 1 − x = e − x = lim e lim x →0 x→0 6 x3 3x 2 6x 1 = 6 1 ex −1− x − x2 1 2 lim = 3 x →0 6 x lim x →1 # & # & x sin x x cos x + sin x − lim % m ( = − lxi→0 % ( x→0 $ x cos x + sin x ' $ − x sin x + cos x + cos x ' 0+0 =− =0 0 +1+1 1 ⎞ྏ ⎛ྎ lim ⎜ྎ cot x − ⎟ྏ = 0 x → 0⎝ྎ...
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