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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^4-2x^2+10
-
x^3+3*x^2-9*x
-
x^3-6*x+cos(x)+sin(x)
-
-x^3+6x^2
-
Expresiones idénticas
- cos(x)/ dieciocho +cos(x*sqrt(diecinueve))+sin(x*sqrt(diecinueve))
- coseno de (x) dividir por 18 más coseno de (x multiplicar por raíz cuadrada de (19)) más seno de (x multiplicar por raíz cuadrada de (19))
- coseno de (x) dividir por dieciocho más coseno de (x multiplicar por raíz cuadrada de (diecinueve)) más seno de (x multiplicar por raíz cuadrada de (diecinueve))
- cos(x)/18+cos(x*√(19))+sin(x*√(19))
- cos(x)/18+cos(xsqrt(19))+sin(xsqrt(19))
- cosx/18+cosxsqrt19+sinxsqrt19
- cos(x) dividir por 18+cos(x*sqrt(19))+sin(x*sqrt(19))
-
Expresiones semejantes
- cos(x)/18-cos(x*sqrt(19))+sin(x*sqrt(19))
- cos(x)/18+cos(x*sqrt(19))-sin(x*sqrt(19))
- cosx/18+cos(x*sqrt(19))+sin(x*sqrt(19))
-
Expresiones con funciones
- Coseno cos
- cos(sqrt(x^2))
- cos(x):|cos(x)|
- cos(2*x-pi/3)*(-3)
- cos(2π*20.4*t+6.6*sin2πt)
- cos(x-pi)-2
- Coseno cos
- cos(sqrt(x^2))
- cos(x):|cos(x)|
- cos(2*x-pi/3)*(-3)
- cos(2π*20.4*t+6.6*sin2πt)
- cos(x-pi)-2
- Raíz cuadrada sqrt
- sqrt(1)*(10-x^2)
- sqrt2/(x+3)
- sqrt(18-6x)
- sqrt(-2*x-1)
- sqrt[5-x]+sqrt[5+x]
- Seno sin
- sin(-x-(pi/3))+1
- sin(x+2)/(x^2-6*x-16)
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- Raíz cuadrada sqrt
- sqrt(1)*(10-x^2)
- sqrt2/(x+3)
- sqrt(18-6x)
- sqrt(-2*x-1)
- sqrt[5-x]+sqrt[5+x]
Gráfico de la función y = cos(x)/18+cos(x*sqrt(19))+sin(x*sqrt(19))
Solución
Ha introducido
[src]
cos(x) / ____\ / ____\
f(x) = ------ + cos\x*\/ 19 / + sin\x*\/ 19 /
18
$$f{\left(x \right)} = \left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}$$
f = cos(x)/18 + cos(sqrt(19)*x) + sin(sqrt(19)*x)
Gráfico de la función
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 50.261957629426$$
$$x_{2} = 45.9500720824419$$
$$x_{3} = -67.9322865974644$$
$$x_{4} = -8.11050797950548$$
$$x_{5} = -29.7311696744772$$
$$x_{6} = -55.6706961951692$$
$$x_{7} = 92.0784360063655$$
$$x_{8} = -34.0624614267879$$
$$x_{9} = 24.3184771377389$$
$$x_{10} = 42.3423105011817$$
$$x_{11} = -78.0113231567737$$
$$x_{12} = -80.1806688839985$$
$$x_{13} = 94.2445646011972$$
$$x_{14} = -73.6935194584558$$
$$x_{15} = 89.9078852723621$$
$$x_{16} = 60.3682235156593$$
$$x_{17} = -52.0707056975749$$
$$x_{18} = -63.6109055061927$$
$$x_{19} = -3.79101548200283$$
$$x_{20} = 74.0571401803626$$
$$x_{21} = -81.6137673091875$$
$$x_{22} = -23.9607984009412$$
$$x_{23} = -11.7178337795199$$
$$x_{24} = 63.9686469875901$$
$$x_{25} = 17.8429060922582$$
$$x_{26} = -60.0093970056837$$
$$x_{27} = -70.08598069901$$
$$x_{28} = 1.97843663832044$$
$$x_{29} = 71.9013470926892$$
$$x_{30} = 78.3705869951396$$
$$x_{31} = 53.8665468375114$$
$$x_{32} = -26.1313710467411$$
$$x_{33} = 16.3896259659427$$
$$x_{34} = 58.1983010519632$$
$$x_{35} = -85.9510076237066$$
$$x_{36} = -88.1182467246423$$
$$x_{37} = -16.0277024282486$$
$$x_{38} = 22.1535757048516$$
$$x_{39} = -65.7755225942878$$
$$x_{40} = 4.1490153050931$$
$$x_{41} = 97.8472894323747$$
$$x_{42} = 56.0289944404762$$
$$x_{43} = -75.8488005014799$$
$$x_{44} = 76.2110844013443$$
$$x_{45} = -41.9787888379248$$
$$x_{46} = 43.0582331960684$$
$$x_{47} = 35.8587565877773$$
$$x_{48} = 34.423829083404$$
$$x_{49} = 68.2953932558856$$
$$x_{50} = 84.1383367630414$$
$$x_{51} = -83.7804112988407$$
$$x_{52} = -13.871703113759$$
$$x_{53} = 30.088339871775$$
$$x_{54} = -21.7932676046496$$
$$x_{55} = -99.6467443289225$$
$$x_{56} = 86.3082811670737$$
$$x_{57} = 9.91798806002494$$
$$x_{58} = 12.0802103138978$$
$$x_{59} = -1.62119029065689$$
$$x_{60} = -39.8251268099257$$
$$x_{61} = 96.4027128688431$$
$$x_{62} = 40.1879016101227$$
$$x_{63} = 19.9965731669714$$
$$x_{64} = -37.6671934984411$$
$$x_{65} = 6.31540430116095$$
$$x_{66} = -44.1358520314859$$
$$x_{67} = 66.1359452473846$$
$$x_{68} = 27.9198639606168$$
$$x_{69} = -5.9545609637231$$
$$x_{70} = 32.2587000138451$$
$$x_{71} = 81.9744966773527$$
$$x_{72} = -45.586523321009$$
$$x_{73} = 100.009210993045$$
$$x_{74} = -47.7410635035814$$
$$x_{75} = 38.027082758287$$
$$x_{76} = -19.633531325068$$
$$x_{77} = -91.7203588600024$$
$$x_{78} = -49.9021808517224$$
$$x_{79} = -57.8411217154807$$
$$x_{80} = 14.2353161195138$$
$$x_{81} = 11.3483955838268$$
$$x_{82} = -26.8485238030702$$
$$x_{83} = -71.5393227898632$$
$$x_{84} = -62.1701414510933$$
$$x_{85} = 48.1037584023775$$
$$x_{86} = -31.9003143336432$$
$$x_{87} = -93.8836077933089$$
$$x_{88} = -96.0393505752796$$
$$x_{89} = 67.5682942687045$$
o sea hay que resolver la ecuación:
$$\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 50.261957629426$$
$$x_{2} = 45.9500720824419$$
$$x_{3} = -67.9322865974644$$
$$x_{4} = -8.11050797950548$$
$$x_{5} = -29.7311696744772$$
$$x_{6} = -55.6706961951692$$
$$x_{7} = 92.0784360063655$$
$$x_{8} = -34.0624614267879$$
$$x_{9} = 24.3184771377389$$
$$x_{10} = 42.3423105011817$$
$$x_{11} = -78.0113231567737$$
$$x_{12} = -80.1806688839985$$
$$x_{13} = 94.2445646011972$$
$$x_{14} = -73.6935194584558$$
$$x_{15} = 89.9078852723621$$
$$x_{16} = 60.3682235156593$$
$$x_{17} = -52.0707056975749$$
$$x_{18} = -63.6109055061927$$
$$x_{19} = -3.79101548200283$$
$$x_{20} = 74.0571401803626$$
$$x_{21} = -81.6137673091875$$
$$x_{22} = -23.9607984009412$$
$$x_{23} = -11.7178337795199$$
$$x_{24} = 63.9686469875901$$
$$x_{25} = 17.8429060922582$$
$$x_{26} = -60.0093970056837$$
$$x_{27} = -70.08598069901$$
$$x_{28} = 1.97843663832044$$
$$x_{29} = 71.9013470926892$$
$$x_{30} = 78.3705869951396$$
$$x_{31} = 53.8665468375114$$
$$x_{32} = -26.1313710467411$$
$$x_{33} = 16.3896259659427$$
$$x_{34} = 58.1983010519632$$
$$x_{35} = -85.9510076237066$$
$$x_{36} = -88.1182467246423$$
$$x_{37} = -16.0277024282486$$
$$x_{38} = 22.1535757048516$$
$$x_{39} = -65.7755225942878$$
$$x_{40} = 4.1490153050931$$
$$x_{41} = 97.8472894323747$$
$$x_{42} = 56.0289944404762$$
$$x_{43} = -75.8488005014799$$
$$x_{44} = 76.2110844013443$$
$$x_{45} = -41.9787888379248$$
$$x_{46} = 43.0582331960684$$
$$x_{47} = 35.8587565877773$$
$$x_{48} = 34.423829083404$$
$$x_{49} = 68.2953932558856$$
$$x_{50} = 84.1383367630414$$
$$x_{51} = -83.7804112988407$$
$$x_{52} = -13.871703113759$$
$$x_{53} = 30.088339871775$$
$$x_{54} = -21.7932676046496$$
$$x_{55} = -99.6467443289225$$
$$x_{56} = 86.3082811670737$$
$$x_{57} = 9.91798806002494$$
$$x_{58} = 12.0802103138978$$
$$x_{59} = -1.62119029065689$$
$$x_{60} = -39.8251268099257$$
$$x_{61} = 96.4027128688431$$
$$x_{62} = 40.1879016101227$$
$$x_{63} = 19.9965731669714$$
$$x_{64} = -37.6671934984411$$
$$x_{65} = 6.31540430116095$$
$$x_{66} = -44.1358520314859$$
$$x_{67} = 66.1359452473846$$
$$x_{68} = 27.9198639606168$$
$$x_{69} = -5.9545609637231$$
$$x_{70} = 32.2587000138451$$
$$x_{71} = 81.9744966773527$$
$$x_{72} = -45.586523321009$$
$$x_{73} = 100.009210993045$$
$$x_{74} = -47.7410635035814$$
$$x_{75} = 38.027082758287$$
$$x_{76} = -19.633531325068$$
$$x_{77} = -91.7203588600024$$
$$x_{78} = -49.9021808517224$$
$$x_{79} = -57.8411217154807$$
$$x_{80} = 14.2353161195138$$
$$x_{81} = 11.3483955838268$$
$$x_{82} = -26.8485238030702$$
$$x_{83} = -71.5393227898632$$
$$x_{84} = -62.1701414510933$$
$$x_{85} = 48.1037584023775$$
$$x_{86} = -31.9003143336432$$
$$x_{87} = -93.8836077933089$$
$$x_{88} = -96.0393505752796$$
$$x_{89} = 67.5682942687045$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$- (19 \sin{\left(\sqrt{19} x \right)} + \frac{\cos{\left(x \right)}}{18} + 19 \cos{\left(\sqrt{19} x \right)}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.8383386969039$$
$$x_{2} = 1.98182013808385$$
$$x_{3} = 86.307552259818$$
$$x_{4} = -35.4962713521321$$
$$x_{5} = 79.8207987925381$$
$$x_{6} = 42.3429010550322$$
$$x_{7} = 35.8564838675336$$
$$x_{8} = 16.3962589987325$$
$$x_{9} = 20.0000857309109$$
$$x_{10} = -83.7847128409657$$
$$x_{11} = 34.4153644903525$$
$$x_{12} = -45.5862378157438$$
$$x_{13} = 53.1543545959115$$
$$x_{14} = -52.0726880597439$$
$$x_{15} = 12.0726583214759$$
$$x_{16} = 30.0903950640639$$
$$x_{17} = 78.3790045638792$$
$$x_{18} = -29.7301992414662$$
$$x_{19} = 92.0736268149492$$
$$x_{20} = -81.6222878012479$$
$$x_{21} = -39.0997253739$$
$$x_{22} = -21.8016403804058$$
$$x_{23} = -44.1442924071731$$
$$x_{24} = 1.26113432597172$$
$$x_{25} = -99.6413295926589$$
$$x_{26} = -91.7133818284164$$
$$x_{27} = 97.8396310051747$$
$$x_{28} = 60.3615745763682$$
$$x_{29} = -47.7480295453976$$
$$x_{30} = 14.2344790722842$$
$$x_{31} = 68.2895634934508$$
$$x_{32} = -63.6048312438269$$
$$x_{33} = -95.3168743179986$$
$$x_{34} = 4.14445714699621$$
$$x_{35} = 48.1085174540459$$
$$x_{36} = 58.1989731458254$$
$$x_{37} = 989.38272092972$$
$$x_{38} = -8.10834067540243$$
$$x_{39} = 100.001805758604$$
$$x_{40} = -31.8927586202166$$
$$x_{41} = -82.3438660914141$$
$$x_{42} = 6.30686856023374$$
$$x_{43} = -70.0908028785674$$
$$x_{44} = 50.2704978681474$$
$$x_{45} = -85.9473508085614$$
$$x_{46} = -39.8206240406073$$
$$x_{47} = -19.6395800198681$$
$$x_{48} = -11.7121867996881$$
$$x_{49} = -93.8756264972675$$
$$x_{50} = -34.0549452295727$$
$$x_{51} = 53.8742026002769$$
$$x_{52} = 81.982673727597$$
$$x_{53} = -55.6761481279815$$
$$x_{54} = 89.9109913224088$$
$$x_{55} = -23.9641129126381$$
$$x_{56} = -1.62162016734916$$
$$x_{57} = 71.8933390918117$$
$$x_{58} = 9.91046775891682$$
$$x_{59} = 32.2530202145542$$
$$x_{60} = -62.1634023800515$$
$$x_{61} = -75.8564896405227$$
$$x_{62} = 96.3980020969127$$
$$x_{63} = 38.0189991331808$$
$$x_{64} = -49.9101620992528$$
$$x_{65} = 66.1275181585432$$
$$x_{66} = 56.036405021783$$
$$x_{67} = -3.78421633182046$$
$$x_{68} = 84.144949641202$$
$$x_{69} = -26.1267495946787$$
$$x_{70} = -5.94647666606874$$
$$x_{71} = -88.1098077014585$$
$$x_{72} = -13.8739441767131$$
$$x_{73} = 74.0551951767426$$
$$x_{74} = 45.9467694361715$$
$$x_{75} = 27.9278717385303$$
$$x_{76} = -37.6586574155057$$
$$x_{77} = -73.6946598879328$$
$$x_{78} = -41.9823708166225$$
$$x_{79} = 62.5238475345719$$
$$x_{80} = 40.1811160542568$$
$$x_{81} = -96.0374799932567$$
$$x_{82} = -78.0186960137561$$
$$x_{83} = -57.8387768251908$$
$$x_{84} = 63.9650580952362$$
$$x_{85} = -60.0012893532383$$
$$x_{86} = -16.0358108793466$$
$$x_{87} = 22.162004186949$$
$$x_{88} = 76.2169563849449$$
$$x_{89} = 24.3243363662851$$
$$x_{90} = -67.9290544628594$$
$$x_{91} = -89.5507926247334$$
$$x_{92} = 94.2360243262519$$
$$x_{93} = -65.7671483275469$$
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[989.38272092972, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -96.0374799932567\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$- (19 \sin{\left(\sqrt{19} x \right)} + \frac{\cos{\left(x \right)}}{18} + 19 \cos{\left(\sqrt{19} x \right)}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.8383386969039$$
$$x_{2} = 1.98182013808385$$
$$x_{3} = 86.307552259818$$
$$x_{4} = -35.4962713521321$$
$$x_{5} = 79.8207987925381$$
$$x_{6} = 42.3429010550322$$
$$x_{7} = 35.8564838675336$$
$$x_{8} = 16.3962589987325$$
$$x_{9} = 20.0000857309109$$
$$x_{10} = -83.7847128409657$$
$$x_{11} = 34.4153644903525$$
$$x_{12} = -45.5862378157438$$
$$x_{13} = 53.1543545959115$$
$$x_{14} = -52.0726880597439$$
$$x_{15} = 12.0726583214759$$
$$x_{16} = 30.0903950640639$$
$$x_{17} = 78.3790045638792$$
$$x_{18} = -29.7301992414662$$
$$x_{19} = 92.0736268149492$$
$$x_{20} = -81.6222878012479$$
$$x_{21} = -39.0997253739$$
$$x_{22} = -21.8016403804058$$
$$x_{23} = -44.1442924071731$$
$$x_{24} = 1.26113432597172$$
$$x_{25} = -99.6413295926589$$
$$x_{26} = -91.7133818284164$$
$$x_{27} = 97.8396310051747$$
$$x_{28} = 60.3615745763682$$
$$x_{29} = -47.7480295453976$$
$$x_{30} = 14.2344790722842$$
$$x_{31} = 68.2895634934508$$
$$x_{32} = -63.6048312438269$$
$$x_{33} = -95.3168743179986$$
$$x_{34} = 4.14445714699621$$
$$x_{35} = 48.1085174540459$$
$$x_{36} = 58.1989731458254$$
$$x_{37} = 989.38272092972$$
$$x_{38} = -8.10834067540243$$
$$x_{39} = 100.001805758604$$
$$x_{40} = -31.8927586202166$$
$$x_{41} = -82.3438660914141$$
$$x_{42} = 6.30686856023374$$
$$x_{43} = -70.0908028785674$$
$$x_{44} = 50.2704978681474$$
$$x_{45} = -85.9473508085614$$
$$x_{46} = -39.8206240406073$$
$$x_{47} = -19.6395800198681$$
$$x_{48} = -11.7121867996881$$
$$x_{49} = -93.8756264972675$$
$$x_{50} = -34.0549452295727$$
$$x_{51} = 53.8742026002769$$
$$x_{52} = 81.982673727597$$
$$x_{53} = -55.6761481279815$$
$$x_{54} = 89.9109913224088$$
$$x_{55} = -23.9641129126381$$
$$x_{56} = -1.62162016734916$$
$$x_{57} = 71.8933390918117$$
$$x_{58} = 9.91046775891682$$
$$x_{59} = 32.2530202145542$$
$$x_{60} = -62.1634023800515$$
$$x_{61} = -75.8564896405227$$
$$x_{62} = 96.3980020969127$$
$$x_{63} = 38.0189991331808$$
$$x_{64} = -49.9101620992528$$
$$x_{65} = 66.1275181585432$$
$$x_{66} = 56.036405021783$$
$$x_{67} = -3.78421633182046$$
$$x_{68} = 84.144949641202$$
$$x_{69} = -26.1267495946787$$
$$x_{70} = -5.94647666606874$$
$$x_{71} = -88.1098077014585$$
$$x_{72} = -13.8739441767131$$
$$x_{73} = 74.0551951767426$$
$$x_{74} = 45.9467694361715$$
$$x_{75} = 27.9278717385303$$
$$x_{76} = -37.6586574155057$$
$$x_{77} = -73.6946598879328$$
$$x_{78} = -41.9823708166225$$
$$x_{79} = 62.5238475345719$$
$$x_{80} = 40.1811160542568$$
$$x_{81} = -96.0374799932567$$
$$x_{82} = -78.0186960137561$$
$$x_{83} = -57.8387768251908$$
$$x_{84} = 63.9650580952362$$
$$x_{85} = -60.0012893532383$$
$$x_{86} = -16.0358108793466$$
$$x_{87} = 22.162004186949$$
$$x_{88} = 76.2169563849449$$
$$x_{89} = 24.3243363662851$$
$$x_{90} = -67.9290544628594$$
$$x_{91} = -89.5507926247334$$
$$x_{92} = 94.2360243262519$$
$$x_{93} = -65.7671483275469$$
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[989.38272092972, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -96.0374799932567\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}\right) = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
$$\lim_{x \to \infty}\left(\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}\right) = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}\right) = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
$$\lim_{x \to \infty}\left(\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}\right) = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)/18 + cos(x*sqrt(19)) + sin(x*sqrt(19)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = - \sin{\left(\sqrt{19} x \right)} + \frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}$$
- No
$$\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = \sin{\left(\sqrt{19} x \right)} - \frac{\cos{\left(x \right)}}{18} - \cos{\left(\sqrt{19} x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = - \sin{\left(\sqrt{19} x \right)} + \frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}$$
- No
$$\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = \sin{\left(\sqrt{19} x \right)} - \frac{\cos{\left(x \right)}}{18} - \cos{\left(\sqrt{19} x \right)}$$
- No
es decir, función
no es
par ni impar