Mister Exam
Graphing y = tan(3*x)/((7*x))
The solution
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tan(3*x)
f(x) = --------
7*x
$$f{\left(x \right)} = \frac{\tan{\left(3 x \right)}}{7 x}$$
f = tan(3*x)/((7*x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\frac{\tan{\left(3 x \right)}}{7 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 30.3687289847013$$
$$x_{2} = -83.7758040957278$$
$$x_{3} = -15.707963267949$$
$$x_{4} = 36.6519142918809$$
$$x_{5} = 76.4454212373516$$
$$x_{6} = -31.4159265358979$$
$$x_{7} = 32.4631240870945$$
$$x_{8} = 74.3510261349584$$
$$x_{9} = 21.9911485751286$$
$$x_{10} = -21.9911485751286$$
$$x_{11} = 4.18879020478639$$
$$x_{12} = 28.2743338823081$$
$$x_{13} = 80.634211442138$$
$$x_{14} = 14.6607657167524$$
$$x_{15} = -96.342174710087$$
$$x_{16} = 72.2566310325652$$
$$x_{17} = 24.0855436775217$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 87.9645943005142$$
$$x_{20} = -50.2654824574367$$
$$x_{21} = -55.5014702134197$$
$$x_{22} = -43.9822971502571$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -97.3893722612836$$
$$x_{25} = 50.2654824574367$$
$$x_{26} = 2.0943951023932$$
$$x_{27} = 59.6902604182061$$
$$x_{28} = -7.33038285837618$$
$$x_{29} = 83.7758040957278$$
$$x_{30} = -61.7846555205993$$
$$x_{31} = 68.0678408277789$$
$$x_{32} = -35.6047167406843$$
$$x_{33} = -41.8879020478639$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = -85.870199198121$$
$$x_{36} = -79.5870138909414$$
$$x_{37} = 12.5663706143592$$
$$x_{38} = -81.6814089933346$$
$$x_{39} = 94.2477796076938$$
$$x_{40} = 52.3598775598299$$
$$x_{41} = 81.6814089933346$$
$$x_{42} = -30.3687289847013$$
$$x_{43} = 54.4542726622231$$
$$x_{44} = 85.870199198121$$
$$x_{45} = -28.2743338823081$$
$$x_{46} = -24.0855436775217$$
$$x_{47} = -48.1710873550435$$
$$x_{48} = -72.2566310325652$$
$$x_{49} = -68.0678408277789$$
$$x_{50} = 96.342174710087$$
$$x_{51} = 37.6991118430775$$
$$x_{52} = -11.5191730631626$$
$$x_{53} = 65.9734457253857$$
$$x_{54} = 39.7935069454707$$
$$x_{55} = -52.3598775598299$$
$$x_{56} = -17.8023583703422$$
$$x_{57} = -87.9645943005142$$
$$x_{58} = 92.1533845053006$$
$$x_{59} = -19.8967534727354$$
$$x_{60} = 61.7846555205993$$
$$x_{61} = 78.5398163397448$$
$$x_{62} = -6.28318530717959$$
$$x_{63} = 15.707963267949$$
$$x_{64} = -57.5958653158129$$
$$x_{65} = -2.0943951023932$$
$$x_{66} = -26.1799387799149$$
$$x_{67} = -70.162235930172$$
$$x_{68} = 56.5486677646163$$
$$x_{69} = 46.0766922526503$$
$$x_{70} = 17.8023583703422$$
$$x_{71} = -90.0589894029074$$
$$x_{72} = 48.1710873550435$$
$$x_{73} = 41.8879020478639$$
$$x_{74} = -65.9734457253857$$
$$x_{75} = 10.471975511966$$
$$x_{76} = 43.9822971502571$$
$$x_{77} = 70.162235930172$$
$$x_{78} = -37.6991118430775$$
$$x_{79} = -59.6902604182061$$
$$x_{80} = -92.1533845053006$$
$$x_{81} = -75.398223686155$$
$$x_{82} = 100.530964914873$$
$$x_{83} = -13.6135681655558$$
$$x_{84} = 34.5575191894877$$
$$x_{85} = 6.28318530717959$$
$$x_{86} = 98.4365698124802$$
$$x_{87} = -99.4837673636768$$
$$x_{88} = -63.8790506229925$$
$$x_{89} = 58.6430628670095$$
$$x_{90} = 19.8967534727354$$
$$x_{91} = -9.42477796076938$$
$$x_{92} = -33.5103216382911$$
$$x_{93} = -4.18879020478639$$
$$x_{94} = 63.8790506229925$$
$$x_{95} = -74.3510261349584$$
$$x_{96} = -46.0766922526503$$
$$x_{97} = 8.37758040957278$$
$$x_{98} = 90.0589894029074$$
$$x_{99} = 26.1799387799149$$
$$x_{100} = -77.4926187885482$$
so we need to solve the equation:
$$\frac{\tan{\left(3 x \right)}}{7 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 30.3687289847013$$
$$x_{2} = -83.7758040957278$$
$$x_{3} = -15.707963267949$$
$$x_{4} = 36.6519142918809$$
$$x_{5} = 76.4454212373516$$
$$x_{6} = -31.4159265358979$$
$$x_{7} = 32.4631240870945$$
$$x_{8} = 74.3510261349584$$
$$x_{9} = 21.9911485751286$$
$$x_{10} = -21.9911485751286$$
$$x_{11} = 4.18879020478639$$
$$x_{12} = 28.2743338823081$$
$$x_{13} = 80.634211442138$$
$$x_{14} = 14.6607657167524$$
$$x_{15} = -96.342174710087$$
$$x_{16} = 72.2566310325652$$
$$x_{17} = 24.0855436775217$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 87.9645943005142$$
$$x_{20} = -50.2654824574367$$
$$x_{21} = -55.5014702134197$$
$$x_{22} = -43.9822971502571$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -97.3893722612836$$
$$x_{25} = 50.2654824574367$$
$$x_{26} = 2.0943951023932$$
$$x_{27} = 59.6902604182061$$
$$x_{28} = -7.33038285837618$$
$$x_{29} = 83.7758040957278$$
$$x_{30} = -61.7846555205993$$
$$x_{31} = 68.0678408277789$$
$$x_{32} = -35.6047167406843$$
$$x_{33} = -41.8879020478639$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = -85.870199198121$$
$$x_{36} = -79.5870138909414$$
$$x_{37} = 12.5663706143592$$
$$x_{38} = -81.6814089933346$$
$$x_{39} = 94.2477796076938$$
$$x_{40} = 52.3598775598299$$
$$x_{41} = 81.6814089933346$$
$$x_{42} = -30.3687289847013$$
$$x_{43} = 54.4542726622231$$
$$x_{44} = 85.870199198121$$
$$x_{45} = -28.2743338823081$$
$$x_{46} = -24.0855436775217$$
$$x_{47} = -48.1710873550435$$
$$x_{48} = -72.2566310325652$$
$$x_{49} = -68.0678408277789$$
$$x_{50} = 96.342174710087$$
$$x_{51} = 37.6991118430775$$
$$x_{52} = -11.5191730631626$$
$$x_{53} = 65.9734457253857$$
$$x_{54} = 39.7935069454707$$
$$x_{55} = -52.3598775598299$$
$$x_{56} = -17.8023583703422$$
$$x_{57} = -87.9645943005142$$
$$x_{58} = 92.1533845053006$$
$$x_{59} = -19.8967534727354$$
$$x_{60} = 61.7846555205993$$
$$x_{61} = 78.5398163397448$$
$$x_{62} = -6.28318530717959$$
$$x_{63} = 15.707963267949$$
$$x_{64} = -57.5958653158129$$
$$x_{65} = -2.0943951023932$$
$$x_{66} = -26.1799387799149$$
$$x_{67} = -70.162235930172$$
$$x_{68} = 56.5486677646163$$
$$x_{69} = 46.0766922526503$$
$$x_{70} = 17.8023583703422$$
$$x_{71} = -90.0589894029074$$
$$x_{72} = 48.1710873550435$$
$$x_{73} = 41.8879020478639$$
$$x_{74} = -65.9734457253857$$
$$x_{75} = 10.471975511966$$
$$x_{76} = 43.9822971502571$$
$$x_{77} = 70.162235930172$$
$$x_{78} = -37.6991118430775$$
$$x_{79} = -59.6902604182061$$
$$x_{80} = -92.1533845053006$$
$$x_{81} = -75.398223686155$$
$$x_{82} = 100.530964914873$$
$$x_{83} = -13.6135681655558$$
$$x_{84} = 34.5575191894877$$
$$x_{85} = 6.28318530717959$$
$$x_{86} = 98.4365698124802$$
$$x_{87} = -99.4837673636768$$
$$x_{88} = -63.8790506229925$$
$$x_{89} = 58.6430628670095$$
$$x_{90} = 19.8967534727354$$
$$x_{91} = -9.42477796076938$$
$$x_{92} = -33.5103216382911$$
$$x_{93} = -4.18879020478639$$
$$x_{94} = 63.8790506229925$$
$$x_{95} = -74.3510261349584$$
$$x_{96} = -46.0766922526503$$
$$x_{97} = 8.37758040957278$$
$$x_{98} = 90.0589894029074$$
$$x_{99} = 26.1799387799149$$
$$x_{100} = -77.4926187885482$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{7 x} \left(3 \tan^{2}{\left(3 x \right)} + 3\right) - \frac{\tan{\left(3 x \right)}}{7 x^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{7 x} \left(3 \tan^{2}{\left(3 x \right)} + 3\right) - \frac{\tan{\left(3 x \right)}}{7 x^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 \left(9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}\right)}{7 x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 70.1638194780521$$
$$x_{2} = 39.7962986804533$$
$$x_{3} = 80.6355893520718$$
$$x_{4} = 85.8714930952558$$
$$x_{5} = 74.3525204774901$$
$$x_{6} = -26.1841813074948$$
$$x_{7} = 46.0791033967769$$
$$x_{8} = -61.7864537594165$$
$$x_{9} = -39.7962986804533$$
$$x_{10} = 94.2489584987906$$
$$x_{11} = 8.39079467531438$$
$$x_{12} = -68.0694730948384$$
$$x_{13} = 58.6449574258964$$
$$x_{14} = -55.5034719939252$$
$$x_{15} = -53.4091553787305$$
$$x_{16} = -30.3723866910662$$
$$x_{17} = -96.343327974564$$
$$x_{18} = 36.654945229763$$
$$x_{19} = -13.6217185706789$$
$$x_{20} = -81.6827692391381$$
$$x_{21} = -63.8807899107774$$
$$x_{22} = -19.9023342041602$$
$$x_{23} = 81.6827692391381$$
$$x_{24} = 12.5751980850599$$
$$x_{25} = -4.21493490101902$$
$$x_{26} = 65.9751298043699$$
$$x_{27} = -37.7020586193797$$
$$x_{28} = 96.343327974564$$
$$x_{29} = -24.0901548043733$$
$$x_{30} = 30.3723866910662$$
$$x_{31} = 92.1545901877593$$
$$x_{32} = 87.9658573926638$$
$$x_{33} = 78.5412309908953$$
$$x_{34} = -94.2489584987906$$
$$x_{35} = 90.060223122721$$
$$x_{36} = -46.0791033967769$$
$$x_{37} = 15.7150294035936$$
$$x_{38} = -85.8714930952558$$
$$x_{39} = 100.532070129183$$
$$x_{40} = 34.5607337431424$$
$$x_{41} = 52.3619994250983$$
$$x_{42} = -11.5288000418182$$
$$x_{43} = -17.8085946429789$$
$$x_{44} = -15.7150294035936$$
$$x_{45} = 48.1733936907667$$
$$x_{46} = 59.6921217440971$$
$$x_{47} = 17.8085946429789$$
$$x_{48} = 28.2782623603624$$
$$x_{49} = -79.5884099297947$$
$$x_{50} = -77.4940525549667$$
$$x_{51} = -48.1733936907667$$
$$x_{52} = -43.9848230807054$$
$$x_{53} = 10.4825608710591$$
$$x_{54} = 19.9023342041602$$
$$x_{55} = -72.2581686851326$$
$$x_{56} = -97.3905131257488$$
$$x_{57} = -9.43653299924975$$
$$x_{58} = -28.2782623603624$$
$$x_{59} = -74.3525204774901$$
$$x_{60} = 2.14462535489162$$
$$x_{61} = 50.267692715982$$
$$x_{62} = 41.890554238415$$
$$x_{63} = 98.4376985407423$$
$$x_{64} = 26.1841813074948$$
$$x_{65} = -41.890554238415$$
$$x_{66} = -50.267692715982$$
$$x_{67} = -65.9751298043699$$
$$x_{68} = 32.4665459324263$$
$$x_{69} = -2.14462535489162$$
$$x_{70} = 83.7771303379305$$
$$x_{71} = 4.21493490101902$$
$$x_{72} = 24.0901548043733$$
$$x_{73} = 21.9961984071905$$
$$x_{74} = -59.6921217440971$$
$$x_{75} = 76.4468746426753$$
$$x_{76} = 63.8807899107774$$
$$x_{77} = -52.3619994250983$$
$$x_{78} = -90.060223122721$$
$$x_{79} = -31.4194623838264$$
$$x_{80} = -21.9961984071905$$
$$x_{81} = -75.3996972758354$$
$$x_{82} = 61.7864537594165$$
$$x_{83} = 72.2581686851326$$
$$x_{84} = -33.513636601095$$
$$x_{85} = -92.1545901877593$$
$$x_{86} = -6.30075451006221$$
$$x_{87} = 54.4563129318412$$
$$x_{88} = -87.9658573926638$$
$$x_{89} = -99.4848842112145$$
$$x_{90} = -70.1638194780521$$
$$x_{91} = -57.59779431589$$
$$x_{92} = 43.9848230807054$$
$$x_{93} = 6.30075451006221$$
$$x_{94} = 14.6683354032458$$
$$x_{95} = -35.6078367878772$$
$$x_{96} = -8.39079467531438$$
$$x_{97} = 68.0694730948384$$
$$x_{98} = 56.5506324811787$$
$$x_{99} = 37.7020586193797$$
$$x_{100} = -83.7771303379305$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}\right)}{7 x}\right) = \frac{18}{7}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}\right)}{7 x}\right) = \frac{18}{7}$$
- limits are equal, then skip the corresponding point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[100.532070129183, \infty\right)$$
Convex at the intervals
$$\left[-2.14462535489162, 2.14462535489162\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 \left(9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}\right)}{7 x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 70.1638194780521$$
$$x_{2} = 39.7962986804533$$
$$x_{3} = 80.6355893520718$$
$$x_{4} = 85.8714930952558$$
$$x_{5} = 74.3525204774901$$
$$x_{6} = -26.1841813074948$$
$$x_{7} = 46.0791033967769$$
$$x_{8} = -61.7864537594165$$
$$x_{9} = -39.7962986804533$$
$$x_{10} = 94.2489584987906$$
$$x_{11} = 8.39079467531438$$
$$x_{12} = -68.0694730948384$$
$$x_{13} = 58.6449574258964$$
$$x_{14} = -55.5034719939252$$
$$x_{15} = -53.4091553787305$$
$$x_{16} = -30.3723866910662$$
$$x_{17} = -96.343327974564$$
$$x_{18} = 36.654945229763$$
$$x_{19} = -13.6217185706789$$
$$x_{20} = -81.6827692391381$$
$$x_{21} = -63.8807899107774$$
$$x_{22} = -19.9023342041602$$
$$x_{23} = 81.6827692391381$$
$$x_{24} = 12.5751980850599$$
$$x_{25} = -4.21493490101902$$
$$x_{26} = 65.9751298043699$$
$$x_{27} = -37.7020586193797$$
$$x_{28} = 96.343327974564$$
$$x_{29} = -24.0901548043733$$
$$x_{30} = 30.3723866910662$$
$$x_{31} = 92.1545901877593$$
$$x_{32} = 87.9658573926638$$
$$x_{33} = 78.5412309908953$$
$$x_{34} = -94.2489584987906$$
$$x_{35} = 90.060223122721$$
$$x_{36} = -46.0791033967769$$
$$x_{37} = 15.7150294035936$$
$$x_{38} = -85.8714930952558$$
$$x_{39} = 100.532070129183$$
$$x_{40} = 34.5607337431424$$
$$x_{41} = 52.3619994250983$$
$$x_{42} = -11.5288000418182$$
$$x_{43} = -17.8085946429789$$
$$x_{44} = -15.7150294035936$$
$$x_{45} = 48.1733936907667$$
$$x_{46} = 59.6921217440971$$
$$x_{47} = 17.8085946429789$$
$$x_{48} = 28.2782623603624$$
$$x_{49} = -79.5884099297947$$
$$x_{50} = -77.4940525549667$$
$$x_{51} = -48.1733936907667$$
$$x_{52} = -43.9848230807054$$
$$x_{53} = 10.4825608710591$$
$$x_{54} = 19.9023342041602$$
$$x_{55} = -72.2581686851326$$
$$x_{56} = -97.3905131257488$$
$$x_{57} = -9.43653299924975$$
$$x_{58} = -28.2782623603624$$
$$x_{59} = -74.3525204774901$$
$$x_{60} = 2.14462535489162$$
$$x_{61} = 50.267692715982$$
$$x_{62} = 41.890554238415$$
$$x_{63} = 98.4376985407423$$
$$x_{64} = 26.1841813074948$$
$$x_{65} = -41.890554238415$$
$$x_{66} = -50.267692715982$$
$$x_{67} = -65.9751298043699$$
$$x_{68} = 32.4665459324263$$
$$x_{69} = -2.14462535489162$$
$$x_{70} = 83.7771303379305$$
$$x_{71} = 4.21493490101902$$
$$x_{72} = 24.0901548043733$$
$$x_{73} = 21.9961984071905$$
$$x_{74} = -59.6921217440971$$
$$x_{75} = 76.4468746426753$$
$$x_{76} = 63.8807899107774$$
$$x_{77} = -52.3619994250983$$
$$x_{78} = -90.060223122721$$
$$x_{79} = -31.4194623838264$$
$$x_{80} = -21.9961984071905$$
$$x_{81} = -75.3996972758354$$
$$x_{82} = 61.7864537594165$$
$$x_{83} = 72.2581686851326$$
$$x_{84} = -33.513636601095$$
$$x_{85} = -92.1545901877593$$
$$x_{86} = -6.30075451006221$$
$$x_{87} = 54.4563129318412$$
$$x_{88} = -87.9658573926638$$
$$x_{89} = -99.4848842112145$$
$$x_{90} = -70.1638194780521$$
$$x_{91} = -57.59779431589$$
$$x_{92} = 43.9848230807054$$
$$x_{93} = 6.30075451006221$$
$$x_{94} = 14.6683354032458$$
$$x_{95} = -35.6078367878772$$
$$x_{96} = -8.39079467531438$$
$$x_{97} = 68.0694730948384$$
$$x_{98} = 56.5506324811787$$
$$x_{99} = 37.7020586193797$$
$$x_{100} = -83.7771303379305$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}\right)}{7 x}\right) = \frac{18}{7}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}\right)}{7 x}\right) = \frac{18}{7}$$
- limits are equal, then skip the corresponding point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[100.532070129183, \infty\right)$$
Convex at the intervals
$$\left[-2.14462535489162, 2.14462535489162\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(3*x)/((7*x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{7 x} \tan{\left(3 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{7 x} \tan{\left(3 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{7 x} \tan{\left(3 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{7 x} \tan{\left(3 x \right)}}{x}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\frac{\tan{\left(3 x \right)}}{7 x} = \frac{\tan{\left(3 x \right)}}{7 x}$$
- No
$$\frac{\tan{\left(3 x \right)}}{7 x} = - \frac{\tan{\left(3 x \right)}}{7 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\tan{\left(3 x \right)}}{7 x} = \frac{\tan{\left(3 x \right)}}{7 x}$$
- No
$$\frac{\tan{\left(3 x \right)}}{7 x} = - \frac{\tan{\left(3 x \right)}}{7 x}$$
- No
so, the function
not is
neither even, nor odd