Calculating Limits of Trigonometric Functions
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EjerciciosLimites trigonometricos
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Calculating Limits of Trigonometric Functions
- 1. 32) 𝐥𝐢𝐦 𝒙→𝟎 𝒔𝒆𝒏 𝟒𝒙 𝟐𝒙 = 𝑠𝑒𝑛 4(0) 2(0) = (0) (0) → 𝐼𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑑𝑜 lim 𝑥→0 𝑠𝑒𝑛 4𝑥 4𝑥 .4𝑥 2𝑥 = lim 𝑥→0 ( 𝑠𝑒𝑛 4𝑥 4𝑥 .4𝑥) lim 𝑥→0 2𝑥 = ( lim 𝑥→0 𝑠𝑒𝑛 4𝑥 4𝑥 ). lim 𝑥→0 (4𝑥) lim 𝑥→0 2𝑥 = (1) lim 𝑥→0 (4𝑥) lim 𝑥→0 2𝑥 = lim 𝑥→0 ( 4𝑥 2𝑥 ) = lim 𝑥→0 2 = 2 Respuesta 33) 𝐥𝐢𝐦 𝒙→𝟎 𝒔𝒆𝒏 𝟐𝒙 𝟏 𝟐 𝒙 = 𝑠𝑒𝑛 2(0) 1 2 (0) = 𝑠𝑒𝑛 0 0 = 0 0 → 𝐼𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑑𝑜 lim 𝑥→0 𝑠𝑒𝑛 2𝑥 2𝑥 .2𝑥 1 2 𝑥 = lim 𝑥→0 ( 𝑠𝑒𝑛 2𝑥 2𝑥 .2𝑥) lim 𝑥→0 1 2 𝑥 = ( lim 𝑥→0 𝑠𝑒𝑛 2𝑥 2𝑥 ). lim 𝑥→0 (2𝑥) lim 𝑥→0 1 2 𝑥 = (1) lim 𝑥→0 (2𝑥) lim 𝑥→0 1 2 𝑥 = lim 𝑥→0 ( 2𝑥 1 2 𝑥 ) = lim 𝑥→0 (4) = 4 Respuesta
- 2. 34) 𝐥𝐢𝐦 𝒙→𝟎 𝒕𝒂𝒏 𝒙 𝟐𝒙 = 𝑡𝑎𝑛 0 2(0) = 0 0 → 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑑𝑎 lim 𝑥→0 𝑠𝑒𝑛 𝑥 cos 𝑥 2𝑥 = lim 𝑥→0 𝑠𝑒𝑛 𝑥 2𝑥𝑐𝑜𝑠 𝑥 = lim 𝑥→0 𝑠𝑒𝑛 𝑥 𝑥 . 𝑥 2𝑥𝑐𝑜𝑠 𝑥 = = lim 𝑥→0 ( 𝑠𝑒𝑛 𝑥 𝑥 .𝑥) lim 𝑥→0 2𝑥𝑐𝑜𝑠 𝑥 = lim 𝑥→0 ( 𝑠𝑒𝑛 𝑥 𝑥 )( lim 𝑥→0 𝑥) (lim 𝑥→0 2𝑥)(lim 𝑥→0 𝑐𝑜𝑠 𝑥) = ( lim 𝑥→0 𝑥) (lim 𝑥→0 2𝑥) = = lim 𝑥→0 𝑥 2𝑥 = lim 1 2 𝑥→0 = ½ Respuesta 35) 𝐥𝐢𝐦 𝒙→𝟎 𝟑𝒙 𝟐 𝟔 𝒔𝒆𝒏 𝟐 𝒙 = 3(0)2 6 𝑠𝑒𝑛20 = 0 0 → 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑐𝑖𝑜𝑛 lim 𝑥→0 3𝑥2 6 𝑠𝑒𝑛2 𝑥 = lim 𝑥→0 3𝑥2 6 𝑠𝑒𝑛 𝑥.𝑠𝑒𝑛 𝑥 = lim 𝑥→0 3𝑥2 6 𝑠𝑒𝑛 𝑥 𝑥 .𝑥 𝑠𝑒𝑛 𝑥 𝑥 .𝑥 = lim 𝑥→0 (3𝑥2) lim 𝑥→0 (6 𝑠𝑒𝑛 𝑥 𝑥 .𝑥 𝑠𝑒𝑛 𝑥 𝑥 .𝑥) = lim 𝑥→0 (3𝑥2) lim 𝑥→0 (6 𝑠𝑒𝑛 𝑥 𝑥 .𝑥 )lim 𝑥→0 (6 𝑠𝑒𝑛 𝑥 𝑥 .𝑥 ) = lim 𝑥→0 (3𝑥2) (lim 𝑥→0 6 ) (lim 𝑥→0 𝑠𝑒𝑛 𝑥 𝑥 )(lim 𝑥→0 𝑥) (lim 𝑥→0 𝑠𝑒𝑛 𝑥 𝑥 )(lim 𝑥→0 𝑥) = lim 𝑥→0 (3𝑥2) lim 𝑥→0 (6)(lim 𝑥→0 𝑥)(lim 𝑥→0 𝑥) = lim 𝑥→0 (3𝑥2) lim 𝑥→0 (6)(lim 𝑥→0 𝑥2) = 6 lim 𝑥→0 (3𝑥2) (lim 𝑥→0 𝑥2) =
- 3. 6 lim 𝑥→0 (3𝑥2) 𝑥2 = 6 (3) = 18 Respuesta