График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 0.2 -0.2
Область определения функции
Точки, в которых функция точно неопределена:x 1 = 0 x_{1} = 0 x 1 = 0
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 ∣ x ∣ sin ( x ) = 0 \frac{1}{\left|{x}\right|} \sin{\left (x \right )} = 0 ∣ x ∣ 1 sin ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π x_{1} = \pi x 1 = π Численное решение x 1 = − 94.2477796077 x_{1} = -94.2477796077 x 1 = − 94.2477796077 x 2 = 31.4159265359 x_{2} = 31.4159265359 x 2 = 31.4159265359 x 3 = 81.6814089933 x_{3} = 81.6814089933 x 3 = 81.6814089933 x 4 = 84.8230016469 x_{4} = 84.8230016469 x 4 = 84.8230016469 x 5 = − 53.407075111 x_{5} = -53.407075111 x 5 = − 53.407075111 x 6 = 65.9734457254 x_{6} = 65.9734457254 x 6 = 65.9734457254 x 7 = 3.14159265359 x_{7} = 3.14159265359 x 7 = 3.14159265359 x 8 = 15.7079632679 x_{8} = 15.7079632679 x 8 = 15.7079632679 x 9 = 100.530964915 x_{9} = 100.530964915 x 9 = 100.530964915 x 10 = 50.2654824574 x_{10} = 50.2654824574 x 10 = 50.2654824574 x 11 = − 3.14159265359 x_{11} = -3.14159265359 x 11 = − 3.14159265359 x 12 = 40.8407044967 x_{12} = 40.8407044967 x 12 = 40.8407044967 x 13 = − 59.6902604182 x_{13} = -59.6902604182 x 13 = − 59.6902604182 x 14 = 97.3893722613 x_{14} = 97.3893722613 x 14 = 97.3893722613 x 15 = 78.5398163397 x_{15} = 78.5398163397 x 15 = 78.5398163397 x 16 = − 25.1327412287 x_{16} = -25.1327412287 x 16 = − 25.1327412287 x 17 = − 43.9822971503 x_{17} = -43.9822971503 x 17 = − 43.9822971503 x 18 = 25.1327412287 x_{18} = 25.1327412287 x 18 = 25.1327412287 x 19 = − 81.6814089933 x_{19} = -81.6814089933 x 19 = − 81.6814089933 x 20 = − 91.1061869541 x_{20} = -91.1061869541 x 20 = − 91.1061869541 x 21 = 87.9645943005 x_{21} = 87.9645943005 x 21 = 87.9645943005 x 22 = 69.115038379 x_{22} = 69.115038379 x 22 = 69.115038379 x 23 = − 34.5575191895 x_{23} = -34.5575191895 x 23 = − 34.5575191895 x 24 = 28.2743338823 x_{24} = 28.2743338823 x 24 = 28.2743338823 x 25 = − 31.4159265359 x_{25} = -31.4159265359 x 25 = − 31.4159265359 x 26 = 37.6991118431 x_{26} = 37.6991118431 x 26 = 37.6991118431 x 27 = − 28.2743338823 x_{27} = -28.2743338823 x 27 = − 28.2743338823 x 28 = 72.2566310326 x_{28} = 72.2566310326 x 28 = 72.2566310326 x 29 = 56.5486677646 x_{29} = 56.5486677646 x 29 = 56.5486677646 x 30 = − 75.3982236862 x_{30} = -75.3982236862 x 30 = − 75.3982236862 x 31 = − 69.115038379 x_{31} = -69.115038379 x 31 = − 69.115038379 x 32 = − 6.28318530718 x_{32} = -6.28318530718 x 32 = − 6.28318530718 x 33 = − 9.42477796077 x_{33} = -9.42477796077 x 33 = − 9.42477796077 x 34 = 6.28318530718 x_{34} = 6.28318530718 x 34 = 6.28318530718 x 35 = 75.3982236862 x_{35} = 75.3982236862 x 35 = 75.3982236862 x 36 = − 65.9734457254 x_{36} = -65.9734457254 x 36 = − 65.9734457254 x 37 = − 87.9645943005 x_{37} = -87.9645943005 x 37 = − 87.9645943005 x 38 = − 72.2566310326 x_{38} = -72.2566310326 x 38 = − 72.2566310326 x 39 = 18.8495559215 x_{39} = 18.8495559215 x 39 = 18.8495559215 x 40 = − 267.035375555 x_{40} = -267.035375555 x 40 = − 267.035375555 x 41 = − 84.8230016469 x_{41} = -84.8230016469 x 41 = − 84.8230016469 x 42 = 9.42477796077 x_{42} = 9.42477796077 x 42 = 9.42477796077 x 43 = − 50.2654824574 x_{43} = -50.2654824574 x 43 = − 50.2654824574 x 44 = − 56.5486677646 x_{44} = -56.5486677646 x 44 = − 56.5486677646 x 45 = − 232.477856366 x_{45} = -232.477856366 x 45 = − 232.477856366 x 46 = − 2642.07942167 x_{46} = -2642.07942167 x 46 = − 2642.07942167 x 47 = 91.1061869541 x_{47} = 91.1061869541 x 47 = 91.1061869541 x 48 = 59.6902604182 x_{48} = 59.6902604182 x 48 = 59.6902604182 x 49 = − 47.1238898038 x_{49} = -47.1238898038 x 49 = − 47.1238898038 x 50 = 12.5663706144 x_{50} = 12.5663706144 x 50 = 12.5663706144 x 51 = − 62.8318530718 x_{51} = -62.8318530718 x 51 = − 62.8318530718 x 52 = 62.8318530718 x_{52} = 62.8318530718 x 52 = 62.8318530718 x 53 = − 18.8495559215 x_{53} = -18.8495559215 x 53 = − 18.8495559215 x 54 = − 12.5663706144 x_{54} = -12.5663706144 x 54 = − 12.5663706144 x 55 = − 37.6991118431 x_{55} = -37.6991118431 x 55 = − 37.6991118431 x 56 = − 97.3893722613 x_{56} = -97.3893722613 x 56 = − 97.3893722613 x 57 = 94.2477796077 x_{57} = 94.2477796077 x 57 = 94.2477796077 x 58 = 34.5575191895 x_{58} = 34.5575191895 x 58 = 34.5575191895 x 59 = − 21.9911485751 x_{59} = -21.9911485751 x 59 = − 21.9911485751 x 60 = 21.9911485751 x_{60} = 21.9911485751 x 60 = 21.9911485751 x 61 = − 100.530964915 x_{61} = -100.530964915 x 61 = − 100.530964915 x 62 = 53.407075111 x_{62} = 53.407075111 x 62 = 53.407075111 x 63 = − 113.097335529 x_{63} = -113.097335529 x 63 = − 113.097335529 x 64 = − 78.5398163397 x_{64} = -78.5398163397 x 64 = − 78.5398163397 x 65 = 0 x_{65} = 0 x 65 = 0 x 66 = 43.9822971503 x_{66} = 43.9822971503 x 66 = 43.9822971503 x 67 = − 40.8407044967 x_{67} = -40.8407044967 x 67 = − 40.8407044967 x 68 = − 15.7079632679 x_{68} = -15.7079632679 x 68 = − 15.7079632679 x 69 = 47.1238898038 x_{69} = 47.1238898038 x 69 = 47.1238898038
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 ∣ 0 ∣ sin ( 0 ) \frac{1}{\left|{0}\right|} \sin{\left (0 \right )} ∣ 0 ∣ 1 sin ( 0 ) f ( 0 ) = N a N f{\left (0 \right )} = \mathrm{NaN} f ( 0 ) = NaN
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 1 ∣ x ∣ sin ( x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{\left|{x}\right|} \sin{\left (x \right )}\right) = 0 x → − ∞ lim ( ∣ x ∣ 1 sin ( x ) ) = 0 Возьмём предел y = 0 y = 0 y = 0 lim x → ∞ ( 1 ∣ x ∣ sin ( x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{\left|{x}\right|} \sin{\left (x \right )}\right) = 0 x → ∞ lim ( ∣ x ∣ 1 sin ( x ) ) = 0 Возьмём предел y = 0 y = 0 y = 0
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(x)/|x|, делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin ( x ) x ∣ x ∣ ) = 0 \lim_{x \to -\infty}\left(\frac{\sin{\left (x \right )}}{x \left|{x}\right|}\right) = 0 x → − ∞ lim ( x ∣ x ∣ sin ( x ) ) = 0 Возьмём предел lim x → ∞ ( sin ( x ) x ∣ x ∣ ) = 0 \lim_{x \to \infty}\left(\frac{\sin{\left (x \right )}}{x \left|{x}\right|}\right) = 0 x → ∞ lim ( x ∣ x ∣ sin ( x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).1 ∣ x ∣ sin ( x ) = − 1 ∣ x ∣ sin ( x ) \frac{1}{\left|{x}\right|} \sin{\left (x \right )} = - \frac{1}{\left|{x}\right|} \sin{\left (x \right )} ∣ x ∣ 1 sin ( x ) = − ∣ x ∣ 1 sin ( x ) 1 ∣ x ∣ sin ( x ) = − 1 ∣ x ∣ ( − 1 sin ( x ) ) \frac{1}{\left|{x}\right|} \sin{\left (x \right )} = - \frac{1}{\left|{x}\right|} \left(-1 \sin{\left (x \right )}\right) ∣ x ∣ 1 sin ( x ) = − ∣ x ∣ 1 ( − 1 sin ( x ) ) 