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2.6 - Review Key - MATH 1205 Review Section 2.6 Evaluate...

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MATH 1205 Review Section: 2.6 Evaluate the limits in exercises 1-9. You must show work to receive credit. 1. lim x + x - 2 + x - 5 x - 1 + x - 1 / 2 = lim x + x - 2 + x - 5 x - 1 + x - 1 / 2 · 1 x - 1 / 2 1 x - 1 / 2 = lim x + x - 2 x - 1 / 2 + x - 5 x - 1 / 2 x - 1 x - 1 / 2 + x - 1 / 2 x - 1 / 2 = lim x + 1 x 3 / 2 + 1 x 9 / 2 1 x 1 / 2 + 1 = lim x + 1 x 3 / 2 + lim x + 1 x 9 / 2 lim x + 1 x 1 / 2 + lim x + 1 = 0 + 0 0 + 1 = 0 2. lim x 7 + x + 7 x 2 - 49 = lim x 7 + x + 7 ( x + 7)( x - 7) = lim x 7 + 1 x - 7 = 1 0 + = + 3. lim x + x 4 + x 2 - x 4 - x 2 (Hint: Rationalize) lim x + x 4 + x 2 - x 4 - x 2 = lim x + x 4 + x 2 - x 4 - x 2 · x 4 + x 2 + x 4 - x 2 x 4 + x 2 + x 4 - x 2 = lim x + ( x 4 + x 2 ) - ( x 4 - x 2 ) x 4 + x 2 + x 4 - x 2 = lim x + 2 x 2 x 4 + x 2 + x 4 - x 2 = lim x + 2 x 2 x 4 + x 2 + x 4 - x 2 · 1 x 2 1 x 2 = lim x + 2 x 2 x 4 + x 2 + x 4 - x 2 · 1 x 2 q 1 x 4 = lim x + 2 x 2 x 2 q x 4 x 4 + x 2 x 4 + q x 4 x 4 - x 2 x 4 = lim x + 2 q 1 + 1 x 2 + q 1 - 1 x 2 = lim x + 2 ( 1 + 1 x 2 ) 1 / 2 + ( 1 - 1 x 2 ) 1 / 2 = lim x + 2 lim x + 1 + lim x + 1 x 2 1 / 2 + lim x + 1 - lim x + 1 x 2 1 / 2 = 2 (1 + 0) 1 / 2 + (1 - 0) 1 / 2 = 2 1 + 1 = 2 2 = 1 . 1
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4. lim x →-∞ 1 x - 3 = lim x →-∞ x 3 = -∞ 5. lim x →-∞ x 2 + 1 x + e Hint: lim x →-∞ x 2 + 1 x + e 6 = lim x + x 2 + 1 x + e ! lim x →-∞ x 2 + 1 x + e = lim x →-∞ x 2 + 1 x + e · 1 | x | 1 | x | = lim x →-∞ x 2 + 1 x + e · q 1 x 2 1 | x | = lim x →-∞ q x 2 x 2 + 1 x 2 x | x | + e | x | = lim x →-∞ q 1 + 1 x 2 x | x | + e | x | = lim x →-∞ ( 1 + 1 x 2 ) 1 / 2 x | x | + e | x | = lim
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