University:
Amherst College
Course:
MATH 121 | Intermediate Calculus
Academic year:

2022
Views:

297

Pages:

2
Author:

Tiffany Zavala

Extra Examples of Limits Math 121 D. Benedetto Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, +∞ or −∞, or Does Not Exist. 0 (0) 1 − 3 sin(3x) − ex 1 − x2 3 + 2x − 3 cosh(3x) 1 + 9x2 √ 0 0 1. lim x→0 arcsin x + cos(3x) − ex ( ) L’H = lim x→0 arctan(3x) + x2 − sinh(3x) 1 x 3 (−2x) − 9 cos(3x) − e 2(1 − x2 ) 2 3 x→0 (18x) + 2 − 9 sinh(3x) − (1 + 9x2 )2 x x 3 − 9 cos(3x) − e 2 −10 −9 − 1 (1 − x ) 2 = lim = = −5 = 54x x→0 2 2 − + 2 − 9 sinh(3x) (1 + 9x2 )2 − L’H = lim 0 (0) 1 + 1 − ( ) ln(1 − x) + x L’H 1−x = lim 2. lim −3x 3 x→0 x→0 cosh(4x) − arctan(3x) − e 4 sinh(4x) − + 3e−3x 1 + 9x2 1 − 1 1 (1 − x)2 L’H =− = − = lim x→0 3(18x) 16 − 9 7 − 9e−3x 16 cosh(4x) + 2 2 (1 + 9x ) 0 0 h 3. lim x→0+ lim ln (1 − 3 sin x) 1 1∞ (1 − 3 sin x) x = ex→0+ 1 x i 1 0 (−3 cos x) ln(1 − 3 sin x) ( 0 ) 1 − 3 sin x lim lim L’H x→0+ x 1 = ex→0+ = e = e−3 3 ∞0 4. lim (ln x) x = lim e x→∞ lim ex→∞ x→∞ 3 ln(ln x) x ∞ ∞ L’H = e lim ln (ln x) = ex→∞   3 1  ln x  x  lim    x→∞ 1 “ 3” ln (ln x) x 3 x 3 lim ln(ln x) = ex→∞ x lim = ex→∞ 3 x ln x = e0 = 1 " 5. lim x→∞ 1 − arcsin 5 x4 3x4 1∞ = e lim ln x→∞ 1 − arcsin 5 x4 3x4 # ( 0 ) 0 5 3 ln 1 − arcsin x4 lim 5 1 x→∞ lim 3x4 ln 1 − arcsin 4 ∞·0 x→∞ x x4 = e =e  ! 3 20 1  − − q 2 x5 1 − arcsin x54 1 − x54 lim 4 x→∞ − 5 L’H x = e  ! 3 1  − q lim 2 (5) x→∞ 1 − arcsin 54 5 x 1 − x4 =e = e−15 x 6 x 1∞ ln 1 − arcsin 6 x = lim e 6. lim 1 − arcsin x→∞ x→∞ x x ∞·0 6 6 lim ln 1 − arcsin lim x ln 1 − arcsin x→∞ x→∞ x x =e =e          1 1 6      − s  2  − x2 ( 0 ) 6  0 6  6 1 − arcsin 1− ln 1 − arcsin x x x lim lim 1 1 x→∞ x→∞ − 2 L’H x x =e = e          1 1 − s  (6)  lim     x→∞ 2 6   6 1 − arcsin 1− x x =e = e−6

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