### Related

? tlimit;

Calculate

##### tlimit

W: %pi*(D^4-x^4)/(32*x);

t: (D-x)/2;

ex1: M/W*tlimit(ex1,x...

Calculate

? tlimit;

Calculate

##### tlimit

W: %pi*(D^4-x^4)/(32*x);

t: (D-x)/2;

ex1: M/W*tlimit(ex1,x...

Calculate

### tlimit

Run Example
```(%i1)tlimit((A * U * (-exp(1/2 * R * t * (-sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V))) + A * U * exp(1/2 * R * t * (sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)) + A * V * exp(1/2 * R * t * (-sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)) - A * V * exp(1/2 * R * t * (sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)) + A * sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) * exp(1/2 * R * t * (-sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)) + A * sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) * exp(1/2 * R * t * (sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)) - 2 * B * Q * exp(1/2 * R * t * (-sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)) + 2 * B * Q * exp(1/2 * R * t * (sqrt(4 * P * Q + U^2 - 2 * U * V + V^2) + U + V)))/(2 * sqrt(4 * P * Q + U^2 - 2 * U * V + V^2)), t, 100000);
2            2
- 50000 R sqrt(V  - 2 U V + U  + 4 P Q)
(%o1) - (%e
2            2                50000 R U
(sqrt(V  - 2 U V + U  + 4 P Q) ((A %e          V
50000 R U
+ (- A U - 2 B Q) %e         )
2            2
100000 R sqrt(V  - 2 U V + U  + 4 P Q) + 50000 R V
%e
50000 R U       50000 R U      50000 R V
+ ((A U + 2 B Q) %e          - A %e          V) %e         )
50000 R U  2           50000 R U           2              50000 R U
+ (- A %e          V  + 2 A U %e          V + (- A U  - 4 A P Q) %e         )
2            2
100000 R sqrt(V  - 2 U V + U  + 4 P Q) + 50000 R V
%e
50000 R U  2           50000 R U           2              50000 R U
+ (- A %e          V  + 2 A U %e          V + (- A U  - 4 A P Q) %e         )
50000 R V       2              2
%e         ))/(2 V  - 4 U V + 2 U  + 8 P Q)
(%i2) ```
Run Example
```tlimit(5*x^4+2*x^2-6,x,5),numer;
(%o1)                               3169.0
(%i2) ```
Run Example
```globalsolve: true;
(%o1)                                true
(%i2) realonly: true;
(%o2)                                true
(%i3) /* Standard (x,y)->
(r,theta) coordinate translation */x(t) := r(t) * cos(theta(t));
(%o3)                     x(t) := r(t) cos(theta(t))
(%i4) y(t) := r(t) * sin(theta(t));
(%o4)                     y(t) := r(t) sin(theta(t))
(%i5) /* Constant course assumption */declare (slope,constant);
(%o5)                                done
(%i6) declare (slope,real);
(%o6)                                done
(%i7) declare (yintercept,constant);
(%o7)                                done
(%i8) declare (yintercept, real);
(%o8)                                done
(%i9) y(t)=slope * x(t) + yintercept;
(%o9)     r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept
(%i10) eq1: y(t)=slope * x(t) + yintercept;
(%o10)    r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept
(%i11) /* Known & constant sound velocity */declare (soundvel,constant);
(%o11)                               done
(%i12) assume(soundvel>
0);
(%o12)                          [0 < soundvel]
(%i13) /* Known and constant base frequency */declare (basefrequency, constant);
(%o13)                               done
(%i14) assume(basefrequency>
0);
(%o14)                        [0 < basefrequency]
(%i15) /* f-- observed frequency. Standard doppler equation*/declare(f, real);
(%o15)                               done
(%i16) let (diff(r(t),t), (f(t)/basefrequency-1)*soundvel);
d                           f(t)
(%o16)            -- (r(t)) --> soundvel (------------- - 1)
dt                      basefrequency
(%i17) diff(r(t),t) = (f(t)/basefrequency-1)*soundvel;
d                         f(t)
(%o17)             -- (r(t)) = soundvel (------------- - 1)
dt                    basefrequency
(%i18) /* Differentiate eq1 and solve for r(t) */eq3: letsimp(solve([diff(y(t),t) = slope * diff(x(t),t)],r(t)));
(%o18) [r(t) =
slope soundvel cos(theta(t))
- -------------------------------------------------------------------
d                               d
slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
dt                              dt
soundvel sin(theta(t))
+ -------------------------------------------------------------------
d                               d
slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
dt                              dt
+ (slope soundvel f(t) cos(theta(t)))/(basefrequency slope sin(theta(t))
d                                             d
(-- (theta(t))) + basefrequency cos(theta(t)) (-- (theta(t))))
dt                                            dt
+ (- (soundvel f(t) sin(theta(t)))/(basefrequency slope sin(theta(t))
d                                             d
(-- (theta(t))) + basefrequency cos(theta(t)) (-- (theta(t)))))]
dt                                            dt
(%i19) /* Differentiate eq3 and solve for slope */eq4: letsimp(diff(eq3,t));
soundvel f(t)
(%o19) [- soundvel + ------------- =
basefrequency
d
soundvel cos(theta(t)) (-- (theta(t)))
dt
-------------------------------------------------------------------
d                               d
slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
dt                              dt
d
slope soundvel sin(theta(t)) (-- (theta(t)))
dt
+ -------------------------------------------------------------------
d                               d
slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
dt                              dt
d
+ (slope soundvel (-- (f(t))) cos(theta(t)))
dt
d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
dt
d
+ basefrequency cos(theta(t)) (-- (theta(t))))
dt
d
+ (- (soundvel (-- (f(t))) sin(theta(t)))
dt
d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
dt
d
+ basefrequency cos(theta(t)) (-- (theta(t)))))
dt
d
+ (- (soundvel f(t) cos(theta(t)) (-- (theta(t))))
dt
d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
dt
d
+ basefrequency cos(theta(t)) (-- (theta(t)))))
dt
d
+ (- (slope soundvel f(t) sin(theta(t)) (-- (theta(t))))
dt
d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
dt
d
+ basefrequency cos(theta(t)) (-- (theta(t)))))
dt
2
d
+ (slope soundvel cos(theta(t)) (slope sin(theta(t)) (--- (theta(t)))
2
dt
2
d                                d             2
+ cos(theta(t)) (--- (theta(t))) - sin(theta(t)) (-- (theta(t)))
2                              dt
dt
d             2
+ slope cos(theta(t)) (-- (theta(t))) ))
dt
d                               d              2
/(slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t))))
dt                              dt
2
d
+ (- (soundvel sin(theta(t)) (slope sin(theta(t)) (--- (theta(t)))
2
dt
2
d                                d             2
+ cos(theta(t)) (--- (theta(t))) - sin(theta(t)) (-- (theta(t)))
2                              dt
dt
d             2
+ slope cos(theta(t)) (-- (theta(t))) ))
dt
d                               d              2
/(slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))) )
dt                              dt
+ (- (slope soundvel f(t) cos(theta(t))
2
d
(basefrequency slope sin(theta(t)) (--- (theta(t)))
2
dt
2
d
+ basefrequency cos(theta(t)) (--- (theta(t)))
2
dt
d             2
- basefrequency sin(theta(t)) (-- (theta(t)))
dt
d             2
+ basefrequency slope cos(theta(t)) (-- (theta(t))) ))
dt
d
/expt(basefrequency slope sin(theta(t)) (-- (theta(t)))
dt
d
+ basefrequency cos(theta(t)) (-- (theta(t))), 2))
dt
+ (soundvel f(t) sin(theta(t)) (basefrequency slope sin(theta(t))
2                                              2
d                                              d
(--- (theta(t))) + basefrequency cos(theta(t)) (--- (theta(t)))
2                                              2
dt                                             dt
d             2
- basefrequency sin(theta(t)) (-- (theta(t)))
dt
d             2
+ basefrequency slope cos(theta(t)) (-- (theta(t))) ))
dt
d
/expt(basefrequency slope sin(theta(t)) (-- (theta(t)))
dt
d
+ basefrequency cos(theta(t)) (-- (theta(t))), 2)]
dt
(%i20) eq6: trigsimp(solve(eq4, slope));
2                                          2
(%o20) [slope = (sqrt((f (t) - 2 basefrequency f(t) + basefrequency )
2
d              2                               d           d
(--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t))) (-- (theta(t)))
2                                            dt          dt
dt
2
d                       2                                             2
(--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t) - 8 basefrequency )
2
dt
d             4    d         2  d             2
(-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
dt                 dt           dt
2
+ ((2 f(t) - 2 basefrequency) sin (theta(t)) - f(t) + basefrequency)
2
d
(--- (theta(t))) + (2 basefrequency - 2 f(t)) cos(theta(t)) sin(theta(t))
2
dt
d             2    d              d             2             d
(-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) sin (theta(t))) (-- (theta(t))))
dt                 dt             dt                          dt
2
d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
2
dt
2
+ ((2 f(t) - 2 basefrequency) sin (theta(t)) + 2 f(t) - 2 basefrequency)
d             2      d
(-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
dt                   dt
d                                  2
(-- (theta(t)))), slope = - (sqrt((f (t) - 2 basefrequency f(t)
dt
2
2   d              2                               d
+ basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
2                                            dt
dt
2
d               d                       2
(-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
dt                2
dt
2   d             4    d         2  d             2
- 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
dt                 dt           dt
2
+ ((2 f(t) - 2 basefrequency) cos (theta(t)) - f(t) + basefrequency)
2
d
(--- (theta(t))) + (2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t))
2
dt
d             2    d              d             2             d
(-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) cos (theta(t))) (-- (theta(t))))
dt                 dt             dt                          dt
2
d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
2
dt
2
+ ((2 basefrequency - 2 f(t)) cos (theta(t)) + 4 f(t) - 4 basefrequency)
d             2      d                                       d
(-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t)) (-- (theta(t))))]
dt                   dt                                      dt
(%i21) /* Now we have a number of estimates for the slope constant. *//* We can use any statistical technique to estimate the actual constant *//* Constant speed assumption */eq7: letsimp(diff(letsimp(diff(x(t),t)),t)) = 0;
2
d
(%o21) (- basefrequency r(t) sin(theta(t)) (--- (theta(t)))
2
dt
d             2
- basefrequency r(t) cos(theta(t)) (-- (theta(t)))
dt
d
- 2 soundvel f(t) sin(theta(t)) (-- (theta(t)))
dt
d
+ 2 basefrequency soundvel sin(theta(t)) (-- (theta(t)))
dt
d
+ soundvel (-- (f(t))) cos(theta(t)))/basefrequency = 0
dt
(%i22) eq8: solve(eq7, r(t));
(%o22) [r(t) = - ((2 soundvel f(t) - 2 basefrequency soundvel) sin(theta(t))
d                          d
(-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t)))
dt                         dt
2
d
/(basefrequency sin(theta(t)) (--- (theta(t)))
2
dt
d             2
+ basefrequency cos(theta(t)) (-- (theta(t))) )]
dt
(%i23) eq9: subst(eq8, eq1);
(%o23) - (sin(theta(t)) ((2 soundvel f(t) - 2 basefrequency soundvel)
d                          d
sin(theta(t)) (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t))))
dt                         dt
2
d
/(basefrequency sin(theta(t)) (--- (theta(t)))
2
dt
d             2
+ basefrequency cos(theta(t)) (-- (theta(t))) ) =
dt
yintercept - (slope cos(theta(t)) ((2 soundvel f(t) - 2 basefrequency soundvel)
d                          d
sin(theta(t)) (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t))))
dt                         dt
2
d
/(basefrequency sin(theta(t)) (--- (theta(t)))
2
dt
d             2
+ basefrequency cos(theta(t)) (-- (theta(t))) )
dt
(%i24) eq10: solve(eq9, yintercept);
(%o24) [yintercept = - (((2 soundvel f(t) - 2 basefrequency soundvel)
2
sin (theta(t)) + (2 basefrequency slope soundvel - 2 slope soundvel f(t))
d
cos(theta(t)) sin(theta(t))) (-- (theta(t)))
dt
d
- soundvel (-- (f(t))) cos(theta(t)) sin(theta(t))
dt
d             2
+ slope soundvel (-- (f(t))) cos (theta(t)))
dt
2
d
/(basefrequency sin(theta(t)) (--- (theta(t)))
2
dt
d             2
+ basefrequency cos(theta(t)) (-- (theta(t))) )]
dt
(%i25) /* Now we have a number of estimates for the yintercept constant. *//* We can use any statistical technique to estimate the actual constant *//* This gives us r(theta) in terms of slope and y_intercept*/eq_rt: solve(eq1, r(t));
yintercept
(%o25)           [r(t) = -----------------------------------]
sin(theta(t)) - slope cos(theta(t))
(%i26) tlimswitch: true;
(%o26)                               true
(%i27) tlimit (eq6, theta, 0);
2
(%o27) [slope = limit      (sqrt((f (t) - 2 basefrequency f(t)
theta -> 0
2
2   d              2                               d
+ basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
2                                            dt
dt
2
d               d                       2
(-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
dt                2
dt
2   d             4    d         2  d             2
- 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
dt                 dt           dt
2
+ ((2 f(t) - 2 basefrequency) sin (theta(t)) - f(t) + basefrequency)
2
d
(--- (theta(t))) + (2 basefrequency - 2 f(t)) cos(theta(t)) sin(theta(t))
2
dt
d             2    d              d             2             d
(-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) sin (theta(t))) (-- (theta(t))))
dt                 dt             dt                          dt
2
d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
2
dt
2
+ ((2 f(t) - 2 basefrequency) sin (theta(t)) + 2 f(t) - 2 basefrequency)
d             2      d
(-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
dt                   dt
d                                              2
(-- (theta(t)))), slope = limit      (- (sqrt((f (t) - 2 basefrequency f(t)
dt                       theta -> 0
2
2   d              2                               d
+ basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
2                                            dt
dt
2
d               d                       2
(-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
dt                2
dt
2   d             4    d         2  d             2
- 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
dt                 dt           dt
2
+ ((2 f(t) - 2 basefrequency) cos (theta(t)) - f(t) + basefrequency)
2
d
(--- (theta(t))) + (2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t))
2
dt
d             2    d              d             2             d
(-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) cos (theta(t))) (-- (theta(t))))
dt                 dt             dt                          dt
2
d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
2
dt
2
+ ((2 basefrequency - 2 f(t)) cos (theta(t)) + 4 f(t) - 4 basefrequency)
d             2      d
(-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
dt                   dt
d
(-- (theta(t)))))]
dt
(%i28) ```

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