Calculus Derivatives
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Limits
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Limits Examples with Answers
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Revised: 9/12/2008 LIMITS NOTES DAY #2 KNOW THE FOLLOWING THREE THEOREMS: 1. 0 sin lim 1 x? = 2. 0 lim 1 sinx? = 3. 0 1 cos lim 0 x? ? = ?? ??/2 ?/2 ? ?2 ?1 1 2 x y ?? ??/2 ?/2 ? 1 2 3 x y ??/2 ?/2 ?1 1 x y Examples: 1. 0 0 0 sin 3 sin 3 3 sin 3 lim lim lim3 3 3 3x x x x x x x x x? ? ? ? ? ? ?? ? ? =? ? ? ?? ? ? ? 2. 0 0 0 1 cos 7 1 cos 7 7 1 cos 7 lim lim lim 7 0 7 7x x x x x x x x x? ? ? ? ? ?? ? ? ?? ? ? =? ? ? ?? ? ? ? 3. 0 0 0 0 0 0 0 sin 2 tan 2 sin 2 sin 2 2cos 2lim lim lim lim cos 2 cos 2 2 2 sin 2 2 2 lim lim lim 2 cos 2 2 cos 2 cos 2(0) x x x x x x x x x x xx x x x x x x x x x x ? ? ? ? ? ? ? ? ?? ? ? ? ?? ?? ? ? ? ? ? =? ?? ? Try These:
Limits Worksheet with Answers
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Revised: 9/28/2008 Limits Worksheet #1 1. 0 lim 2 7 x x? + 6. 3 2 8lim 2x x x? ? ? 2. 2 3 lim 3 8 x x x? + ? 7. 9 3 lim 9x x x? ? ? 3. 2 5 25lim 5x x x? ? ? 8. 2 33 9lim 27x x x? ? ? 4. 2 2 5 6lim 2x x x x? ? + ? 9. 3 1 7lim 8x x x? + + 5. 22 2lim 4x x x? ? ? 10. 2 2 lim 2 1 x x? + Answers:
Limits to infinity
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If a limit is approaching infinity, the highest exponent on the numerator or denominator determines whether its zero, or infinity
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