График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 -0.10 0.05
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 ( 5 x ) = 0 \frac{1}{x} \sin{\left (5 x \right )} = 0 x 1 sin ( 5 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 5 x_{1} = \frac{\pi}{5} x 1 = 5 π Численное решение x 1 = 52.7787565803 x_{1} = 52.7787565803 x 1 = 52.7787565803 x 2 = − 93.619461077 x_{2} = -93.619461077 x 2 = − 93.619461077 x 3 = 74.1415866247 x_{3} = 74.1415866247 x 3 = 74.1415866247 x 4 = − 96.1327351998 x_{4} = -96.1327351998 x 4 = − 96.1327351998 x 5 = 98.017690792 x_{5} = 98.017690792 x 5 = 98.017690792 x 6 = − 27.6460153516 x_{6} = -27.6460153516 x 6 = − 27.6460153516 x 7 = 10.0530964915 x_{7} = 10.0530964915 x 7 = 10.0530964915 x 8 = − 57.8053048261 x_{8} = -57.8053048261 x 8 = − 57.8053048261 x 9 = − 33.9292006588 x_{9} = -33.9292006588 x 9 = − 33.9292006588 x 10 = 48.3805268653 x_{10} = 48.3805268653 x 10 = 48.3805268653 x 11 = 16.3362817987 x_{11} = 16.3362817987 x 11 = 16.3362817987 x 12 = 54.0353936417 x_{12} = 54.0353936417 x 12 = 54.0353936417 x 13 = 84.1946831162 x_{13} = 84.1946831162 x 13 = 84.1946831162 x 14 = 70.3716754404 x_{14} = 70.3716754404 x 14 = 70.3716754404 x 15 = − 25.7610597594 x_{15} = -25.7610597594 x 15 = − 25.7610597594 x 16 = − 5.65486677646 x_{16} = -5.65486677646 x 16 = − 5.65486677646 x 17 = − 74.1415866247 x_{17} = -74.1415866247 x 17 = − 74.1415866247 x 18 = 64.0884901332 x_{18} = 64.0884901332 x 18 = 64.0884901332 x 19 = 43.9822971503 x_{19} = 43.9822971503 x 19 = 43.9822971503 x 20 = 89.221231362 x_{20} = 89.221231362 x 20 = 89.221231362 x 21 = − 35.8141562509 x_{21} = -35.8141562509 x 21 = − 35.8141562509 x 22 = 18.2212373908 x_{22} = 18.2212373908 x 22 = 18.2212373908 x 23 = 5.65486677646 x_{23} = 5.65486677646 x 23 = 5.65486677646 x 24 = 30.1592894745 x_{24} = 30.1592894745 x 24 = 30.1592894745 x 25 = − 47.7522083346 x_{25} = -47.7522083346 x 25 = − 47.7522083346 x 26 = − 11.9380520836 x_{26} = -11.9380520836 x 26 = − 11.9380520836 x 27 = 20.106192983 x_{27} = 20.106192983 x 27 = 20.106192983 x 28 = − 79.7964534012 x_{28} = -79.7964534012 x 28 = − 79.7964534012 x 29 = − 13.8230076758 x_{29} = -13.8230076758 x 29 = − 13.8230076758 x 30 = 33.9292006588 x_{30} = 33.9292006588 x 30 = 33.9292006588 x 31 = 87.9645943005 x_{31} = 87.9645943005 x 31 = 87.9645943005 x 32 = 40.2123859659 x_{32} = 40.2123859659 x 32 = 40.2123859659 x 33 = − 32.0442450666 x_{33} = -32.0442450666 x 33 = − 32.0442450666 x 34 = − 89.8495498927 x_{34} = -89.8495498927 x 34 = − 89.8495498927 x 35 = 26.3893782902 x_{35} = 26.3893782902 x 35 = 26.3893782902 x 36 = − 20.106192983 x_{36} = -20.106192983 x 36 = − 20.106192983 x 37 = 8.16814089933 x_{37} = 8.16814089933 x 37 = 8.16814089933 x 38 = 82.3097275241 x_{38} = 82.3097275241 x 38 = 82.3097275241 x 39 = − 76.0265422169 x_{39} = -76.0265422169 x 39 = − 76.0265422169 x 40 = 23.8761041673 x_{40} = 23.8761041673 x 40 = 23.8761041673 x 41 = − 21.9911485751 x_{41} = -21.9911485751 x 41 = − 21.9911485751 x 42 = − 86.0796387084 x_{42} = -86.0796387084 x 42 = − 86.0796387084 x 43 = − 37.6991118431 x_{43} = -37.6991118431 x 43 = − 37.6991118431 x 44 = 86.0796387084 x_{44} = 86.0796387084 x 44 = 86.0796387084 x 45 = 21.9911485751 x_{45} = 21.9911485751 x 45 = 21.9911485751 x 46 = 0 x_{46} = 0 x 46 = 0 x 47 = 60.3185789489 x_{47} = 60.3185789489 x 47 = 60.3185789489 x 48 = − 69.7433569097 x_{48} = -69.7433569097 x 48 = − 69.7433569097 x 49 = 38.3274303738 x_{49} = 38.3274303738 x 49 = 38.3274303738 x 50 = 52.1504380496 x_{50} = 52.1504380496 x 50 = 52.1504380496 x 51 = − 91.7345054848 x_{51} = -91.7345054848 x 51 = − 91.7345054848 x 52 = 77.911497809 x_{52} = 77.911497809 x 52 = 77.911497809 x 53 = 50.2654824574 x_{53} = 50.2654824574 x 53 = 50.2654824574 x 54 = − 59.6902604182 x_{54} = -59.6902604182 x 54 = − 59.6902604182 x 55 = 28.2743338823 x_{55} = 28.2743338823 x 55 = 28.2743338823 x 56 = − 43.9822971503 x_{56} = -43.9822971503 x 56 = − 43.9822971503 x 57 = − 81.6814089933 x_{57} = -81.6814089933 x 57 = − 81.6814089933 x 58 = 99.9026463842 x_{58} = 99.9026463842 x 58 = 99.9026463842 x 59 = 72.2566310326 x_{59} = 72.2566310326 x 59 = 72.2566310326 x 60 = − 49.6371639267 x_{60} = -49.6371639267 x 60 = − 49.6371639267 x 61 = − 65.9734457254 x_{61} = -65.9734457254 x 61 = − 65.9734457254 x 62 = − 1.88495559215 x_{62} = -1.88495559215 x 62 = − 1.88495559215 x 63 = 42.0973415581 x_{63} = 42.0973415581 x 63 = 42.0973415581 x 64 = 11.3097335529 x_{64} = 11.3097335529 x 64 = 11.3097335529 x 65 = 189.752196277 x_{65} = 189.752196277 x 65 = 189.752196277 x 66 = − 52.7787565803 x_{66} = -52.7787565803 x 66 = − 52.7787565803 x 67 = 45.8672527424 x_{67} = 45.8672527424 x 67 = 45.8672527424 x 68 = 67.8584013175 x_{68} = 67.8584013175 x 68 = 67.8584013175 x 69 = − 10.0530964915 x_{69} = -10.0530964915 x 69 = − 10.0530964915 x 70 = − 55.9203492339 x_{70} = -55.9203492339 x 70 = − 55.9203492339 x 71 = 94.2477796077 x_{71} = 94.2477796077 x 71 = 94.2477796077 x 72 = − 45.8672527424 x_{72} = -45.8672527424 x 72 = − 45.8672527424 x 73 = − 3.76991118431 x_{73} = -3.76991118431 x 73 = − 3.76991118431 x 74 = − 42.0973415581 x_{74} = -42.0973415581 x 74 = − 42.0973415581 x 75 = − 98.017690792 x_{75} = -98.017690792 x 75 = − 98.017690792 x 76 = − 15.7079632679 x_{76} = -15.7079632679 x 76 = − 15.7079632679 x 77 = 32.0442450666 x_{77} = 32.0442450666 x 77 = 32.0442450666 x 78 = 65.9734457254 x_{78} = 65.9734457254 x 78 = 65.9734457254 x 79 = − 71.6283125018 x_{79} = -71.6283125018 x 79 = − 71.6283125018 x 80 = − 23.8761041673 x_{80} = -23.8761041673 x 80 = − 23.8761041673 x 81 = 55.9203492339 x_{81} = 55.9203492339 x 81 = 55.9203492339 x 82 = − 54.0353936417 x_{82} = -54.0353936417 x 82 = − 54.0353936417 x 83 = 1.88495559215 x_{83} = 1.88495559215 x 83 = 1.88495559215 x 84 = − 67.8584013175 x_{84} = -67.8584013175 x 84 = − 67.8584013175 x 85 = 92.3628240155 x_{85} = 92.3628240155 x 85 = 92.3628240155 x 86 = − 64.0884901332 x_{86} = -64.0884901332 x 86 = − 64.0884901332 x 87 = 6.28318530718 x_{87} = 6.28318530718 x 87 = 6.28318530718 x 88 = − 87.9645943005 x_{88} = -87.9645943005 x 88 = − 87.9645943005 x 89 = 96.1327351998 x_{89} = 96.1327351998 x 89 = 96.1327351998 x 90 = 101.159283446 x_{90} = 101.159283446 x 90 = 101.159283446 x 91 = 11.9380520836 x_{91} = 11.9380520836 x 91 = 11.9380520836 x 92 = − 99.9026463842 x_{92} = -99.9026463842 x 92 = − 99.9026463842 x 93 = 5.02654824574 x_{93} = 5.02654824574 x 93 = 5.02654824574 x 94 = 76.0265422169 x_{94} = 76.0265422169 x 94 = 76.0265422169 x 95 = − 77.911497809 x_{95} = -77.911497809 x 95 = − 77.911497809 x 96 = 62.2035345411 x_{96} = 62.2035345411 x 96 = 62.2035345411
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 0 sin ( 0 ⋅ 5 ) \frac{1}{0} \sin{\left (0 \cdot 5 \right )} 0 1 sin ( 0 ⋅ 5 ) 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 ( 5 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \sin{\left (5 x \right )}\right) = 0 x → − ∞ lim ( x 1 sin ( 5 x ) ) = 0 Возьмём предел y = 0 y = 0 y = 0 lim x → ∞ ( 1 x sin ( 5 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \sin{\left (5 x \right )}\right) = 0 x → ∞ lim ( x 1 sin ( 5 x ) ) = 0 Возьмём предел y = 0 y = 0 y = 0
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(5*x)/x, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x 2 sin ( 5 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x^{2}} \sin{\left (5 x \right )}\right) = 0 x → − ∞ lim ( x 2 1 sin ( 5 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 x 2 sin ( 5 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x^{2}} \sin{\left (5 x \right )}\right) = 0 x → ∞ lim ( x 2 1 sin ( 5 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 ( 5 x ) = 1 x sin ( 5 x ) \frac{1}{x} \sin{\left (5 x \right )} = \frac{1}{x} \sin{\left (5 x \right )} x 1 sin ( 5 x ) = x 1 sin ( 5 x ) 1 x sin ( 5 x ) = − 1 x sin ( 5 x ) \frac{1}{x} \sin{\left (5 x \right )} = - \frac{1}{x} \sin{\left (5 x \right )} x 1 sin ( 5 x ) = − x 1 sin ( 5 x ) 