(%i1) load(draw)$

1 Limes funkcije

Napomena. U svakom zadatku su prije racunanja limesa definirane funkcije zasebno,
kako biste zorno jasnije vidjeli o kojoj se funkciji radi. Naravno da nema potrebe funkciju
zasebno definirati, moze se njezina formula direktno unositi u naredbu "limit".

1. zadatak

(%i2) f(x):=(sqrt(x)+1)/(1-x);
(%o2) f(x):=(1+sqrt(x))/(1-x)
(%i3) limit(f(x),x,5);
(%o3) -(1+sqrt(5))/4

2. zadatak

(%i4) f(x):=(sqrt(1+2*x)-3)/(sqrt(x)-2);
(%o4) f(x):=(sqrt(2*x+1)-3)/(sqrt(x)-2)
(%i5) limit(f(x),x,inf);
(%o5) sqrt(2)

3. zadatak

(%i6) f(x):=(x^(1/2)+x^(1/3)+x^(1/4))/(4*x-3)^(1/2);
(%o6) f(x):=(x^(1/4)+x^(1/3)+x^(1/2))/(4*x-3)^(1/2)
(%i7) limit(f(x),x,inf);
(%o7) 1/2

4. zadatak

(%i8) f(x):=(x+3)/(9-x^2);
(%o8) f(x):=(3+x)/(9-x^2)
(%i9) limit(f(x),x,-3);
(%o9) 1/6
(%i10) wxplot2d([(x+3)/(9-x^2)], [x,-5,6],[y,-10,10])$
plot2d: some values were clipped.
(%t10)
(Graphics)

5. zadatak

(%i11) f(x):=(x^2-x-12)/(x-4);
(%o11) f(x):=(-12-x+x^2)/(x-4)
(%i12) limit(f(x),x,4);
(%o12) 7

6. zadatak

(%i13) f(x):=(x^3-27)/(x^2-9);
(%o13) f(x):=(x^3-27)/(x^2-9)
(%i14) limit(f(x),x,3);
(%o14) 9/2
(%i15) wxplot2d([(x^3-27)/(x^2-9)], [x,-5,6],[y,-2,20])$
plot2d: some values were clipped.
(%t15)
(Graphics)
(%i16) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",line_width=3,color=blue,
xtics=[-4,1,6],ytics=[-2,1,20],explicit((x^3-27)/(x^2-9),x,-5,6),line_width=1.5,color=red,
explicit(4.5,x,-5,3),parametric(3,t,t,-2,4.5),point_type=filled_circle,point_size=1.5,
points([[3,4.5]]),yrange=[-2,20]),wxplot_size=[800,500];
(%t16)
(Graphics)
(%o16)
(%i17) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",line_width=3,color=blue,
xtics=[-4,1,6],ytics=[-2,0.5,20],explicit((x^3-27)/(x^2-9),x,-5,6),line_width=1.5,color=red,
explicit(4.5,x,-5,3),parametric(3,t,t,-2,4.5),point_type=filled_circle,point_size=1.5,
points([[3,4.5]]),yrange=[-2,20]);
(%o17) [gr2d(explicit,explicit,parametric,points)]

7. zadatak

(%i18) f(x):=(x^2+x-2)/(x-1)^2;
(%o18) f(x):=(-2+x+x^2)/(x-1)^2
(%i19) limit(f(x),x,1);
(%o19) infinity

limes sa lijeva

(%i20) limit(f(x),x,1,minus);
(%o20) -inf

limes zdesna

(%i21) limit(f(x),x,1,plus);
(%o21) inf
(%i22) wxplot2d([(x^2+x-2)/(x-1)^2], [x,-5,6],[y,-10,10])$
plot2d: some values were clipped.
(%t22)
(Graphics)

8. zadatak

(%i23) f(x):=(x^3-3*x^2-4*x+12)/(x^2-5*x+6);
(%o23) f(x):=(12-4*x-3*x^2+x^3)/(6-5*x+x^2)
(%i24) limit(f(x),x,3);
(%o24) 5

9. zadatak

(%i25) f(x):=(sqrt(x)-2)/(4-x);
(%o25) f(x):=(sqrt(x)-2)/(4-x)
(%i26) limit(f(x),x,4);
(%o26) -1/4

10. zadatak

(%i27) f(x):=(sqrt(1+2*x)-3)/(sqrt(x)-2);
(%o27) f(x):=(sqrt(2*x+1)-3)/(sqrt(x)-2)
(%i28) limit(f(x),x,4);
(%o28) 4/3

11. zadatak

(%i29) f(x):=sqrt(x-2)/(x^2-4);
(%o29) f(x):=sqrt(x-2)/(x^2-4)
(%i30) limit(f(x),x,2,plus);
(%o30) inf

12. zadatak

(%i31) f(x):=sin(7*x)/x;
(%o31) f(x):=sin(7*x)/x
(%i32) limit(f(x),x,0);
(%o32) 7

13. zadatak

(%i33) f(x):=(1-cos(x))/(2*x^2);
(%o33) f(x):=(1-cos(x))/(2*x^2)
(%i34) limit(f(x),x,0);
(%o34) 1/4

14. zadatak

(%i35) f(x):=tan(%pi*x)/(x+4);
(%o35) f(x):=tan(%pi*x)/(4+x)
(%i36) limit(f(x),x,-4);
(%o36) %pi

15. zadatak

(%i37) f(x):=(cos(2*x)-cos(4))/(14*x+28);
(%o37) f(x):=(cos(2*x)-cos(4))/(28+14*x)
(%i38) limit(f(x),x,-2);
(%o38) (2*cos(2)*sin(2))/7
(%i39) trigreduce(%);
(%o39) sin(4)/7

16. zadatak

(%i40) f(x):=((x+1)/(x-2))^(3*x+1);
(%o40) f(x):=((1+x)/(x-2))^(3*x+1)
(%i41) limit(f(x),x,inf);
(%o41) %e^9

17. zadatak

(%i42) f(x):=(1+1/4*x)^(1/x);
(%o42) f(x):=(1+x/4)^(1/x)
(%i43) limit(f(x),x,0);
(%o43) %e^(1/4)

18. zadatak

(%i44) f(x):=(2^x+5^x)/(5^x-4^x);
(%o44) f(x):=(5^x+2^x)/(5^x-4^x)
(%i45) limit(f(x),x,inf);
(%o45) 1

19. zadatak

(%i46) f(x):=(3^x-12^x)/x;
(%o46) f(x):=(3^x-12^x)/x
(%i47) limit(f(x),x,0);
(%o47) log(3)-log(12)
(%i48) logcontract(%);
(%o48) -log(4)

20. zadatak

(%i49) f(x):=(log(1+x)/log(10)-log(2)/log(10))/(x-1);
(%o49) f(x):=(log(x+1)/log(10)-log(2)/log(10))/(x-1)
(%i50) limit(f(x),x,1);
(%o50) 1/(2*log(10))

21. zadatak

(%i51) f(x):=((x^2+2)/(x^2+5))^((x^2+3)/3);
(%o51) f(x):=((2+x^2)/(5+x^2))^((3+x^2)/3)
(%i52) limit(f(x),x,0);
(%o52) 2/5

22. zadatak

(%i53) f(x):=log(x+2)/(x+1);
(%o53) f(x):=log(2+x)/(1+x)
(%i54) limit(f(x),x,-1);
(%o54) 1

23. zadatak

(%i55) f(x):=(45^(x+1)-1)/(3*x+3);
(%o55) f(x):=(45^(x+1)-1)/(3+3*x)
(%i56) limit(f(x),x,-1);
(%o56) log(45)/3

24. zadatak

(%i57) f(x):=(%e^(x+3)-%e)/(x+2);
(%o57) f(x):=(%e^(x+3)-%e)/(2+x)
(%i58) limit(f(x),x,-2);
(%o58) %e

25. zadatak

(%i59) f(x):=(x-1)/log(x);
(%o59) f(x):=(x-1)/log(x)
(%i60) limit(f(x),x,1);
(%o60) 1

26. zadatak - limes ne postoji (funkcija ne odlazi u beskonacno, ali oscilira)

(%i61) limit(sin(x),x,inf);
(%o61) ind

2 Vrste prekida

1. zadatak (uklonjiv prekid prve vrste)

(%i62) f(x):=(-x^2+5*x-6)/(2-x);
(%o62) f(x):=(-6+5*x-x^2)/(2-x)
(%i63) limit(f(x),x,2,minus);
(%o63) -1
(%i64) limit(f(x),x,2,plus);
(%o64) -1
(%i65) wxplot2d([(-x^2+5*x-6)/(2-x),[discrete,[[2,-1]]]], [x,-2,5],[y,-6,4],grid2d,
[style, [lines,3],[points]],[point_type, bullet],[legend,false])$
(%t65)
(Graphics)

2. zadatak (neuklonjiv prekid prve vrste)

(%i66) f(x):=(x-7)/abs(x-7);
(%o66) f(x):=(x-7)/abs(x-7)
(%i67) limit(f(x),x,7,minus);
(%o67) -1
(%i68) limit(f(x),x,7,plus);
(%o68) 1
(%i69) wxplot2d([(x-7)/abs(x-7),[discrete,[[7,-1],[7,1]]]], [x,-1,10],[y,-6,4],grid2d,
[style, [lines,3],points],[point_type,bullet],[legend,false])$
(%t69)
(Graphics)
(%i70) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-1,1,15],ytics=[-6,1,4],line_width=3,color=blue,
explicit(-1,x,-1,7), explicit(1,x,7,15),color=red,point_type=filled_circle,point_size=1.5,
points([[7,-1],[7,1]]),yrange=[-6,4]),wxplot_size=[600,400];
(%t70)
(Graphics)
(%o70)
(%i71) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-1,1,15],ytics=[-6,1,4],line_width=3,color=blue,
explicit(-1,x,-1,7), explicit(1,x,7,15),color=red,point_type=filled_circle,point_size=1.5,
points([[7,-1],[7,1]]),yrange=[-6,4]);
(%o71) [gr2d(explicit,explicit,points)]

3. zadatak (prekid druge vrste)

(%i72) f(x):=4/(x-1);
(%o72) f(x):=4/(x-1)
(%i73) limit(f(x),x,1,minus);
(%o73) -inf
(%i74) limit(f(x),x,1,plus);
(%o74) inf
(%i75) wxplot2d([4/(x-1)], [x,-1,3],[y,-60,60],grid2d,
[style, [lines,3]])$
plot2d: expression evaluates to non-numeric value somewhere in plotting range.plot2d: some values were clipped.<BR>
(%t75)
(Graphics)

4. zadatak (neuklonjiv prekid prve vrste)

(%i76) f(x):=1/(4+%e^(1/x));
(%o76) f(x):=1/(%e^(1/x)+4)
(%i77) limit(f(x),x,0,minus);
(%o77) 1/4
(%i78) limit(f(x),x,0,plus);
(%o78) 0
(%i79) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,7],ytics=[-0.1,0.1,0.3],
line_width=3,color=blue,explicit(1/(4+%e^(1/x)),x,-3,-0.01),explicit(1/(4+%e^(1/x)),x,0.01,7),
yrange=[-0.2,0.4]),wxplot_size=[600,400];
(%t79)
(Graphics)
(%o79)
(%i80) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,7],ytics=[-0.1,0.1,0.3],
line_width=3,color=blue,explicit(1/(4+%e^(1/x)),x,-3,-0.01),explicit(1/(4+%e^(1/x)),x,0.01,7),
yrange=[-0.2,0.4]);
(%o80) [gr2d(explicit,explicit)]

5. zadatak (neuklonjiv prekid prve vrste)

(%i81) f(x):=sqrt(1+cos(2*x))/(sqrt(%pi)-sqrt(2*x));
(%o81) f(x):=sqrt(cos(2*x)+1)/(sqrt(%pi)-sqrt(2*x))
(%i82) limit(f(x),x,%pi/2,minus);
(%o82) sqrt(2)*sqrt(%pi)
(%i83) limit(f(x),x,%pi/2,plus);
(%o83) -sqrt(2)*sqrt(%pi)
(%i84) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[0,1,7],ytics=[-3,1,3],
line_width=3,color=blue,explicit(f(x),x,0,1.57),explicit(f(x),x,1.58,7),
color=black,line_width=2,line_type=dots,parametric(%pi/2,t,t,-sqrt(2*%pi),sqrt(2*%pi)),
yrange=[-3,3]),wxplot_size=[600,400];
(%t84)
(Graphics)
(%o84)
(%i85) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[0,1,7],ytics=[-3,1,3],
line_width=3,color=blue,explicit(f(x),x,0,1.57),explicit(f(x),x,1.58,7),
color=black,line_width=2,line_type=dots,parametric(%pi/2,t,t,-sqrt(2*%pi),sqrt(2*%pi)),
yrange=[-3,3]);
(%o85) [gr2d(explicit,explicit,parametric)]

6. zadatak (prekid druge vrste)

(%i86) f(x):=%e^(1/x);
(%o86) f(x):=%e^(1/x)
(%i87) limit(f(x),x,0,minus);
(%o87) 0
(%i88) limit(f(x),x,0,plus);
(%o88) inf
(%i89) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,3],ytics=[0,2,20],
line_width=3,color=blue,explicit(f(x),x,-3,-0.01),explicit(f(x),x,0.01,3),
yrange=[-1,20]),wxplot_size=[600,400];
(%t89)
(Graphics)
(%o89)
(%i90) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,3],ytics=[0,2,20],
line_width=3,color=blue,explicit(f(x),x,-3,-0.01),explicit(f(x),x,0.01,3),
yrange=[-1,20]);
(%o90) [gr2d(explicit,explicit)]

7. zadatak (prekid druge vrste)

(%i91) f(x):=%e^(1/x^2);
(%o91) f(x):=%e^(1/x^2)
(%i92) limit(f(x),x,0,minus);
(%o92) inf
(%i93) limit(f(x),x,0,plus);
(%o93) inf
(%i94) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,3],ytics=[0,2,20],
line_width=3,color=blue,explicit(f(x),x,-3,3),
yrange=[-1,20]),wxplot_size=[600,400];
(%t94)
(Graphics)
(%o94)
(%i95) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,3],ytics=[0,2,20],
line_width=3,color=blue,explicit(f(x),x,-3,3),
yrange=[-1,20]);
(%o95) [gr2d(explicit)]

8. zadatak (uklonjiv prekid prve vrste)

(%i96) f(x):=%e^(-1/x^2);
(%o96) f(x):=%e^((-1)/x^2)
(%i97) limit(f(x),x,0,minus);
(%o97) 0
(%i98) limit(f(x),x,0,plus);
(%o98) 0
(%i99) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-5,1,5],ytics=[0,0.1,1],
line_width=3,color=blue,explicit(f(x),x,-5,5),color=red,
point_type=filled_circle,point_size=1.5,points([[0,0]]),
yrange=[-0.1,1]),wxplot_size=[600,400];
(%t99)
(Graphics)
(%o99)
(%i100) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-5,1,5],ytics=[0,0.1,1],
line_width=3,color=blue,explicit(f(x),x,-5,5),color=red,
point_type=filled_circle,point_size=1.5,points([[0,0]]),
yrange=[-0.1,1]),wxplot_size=[600,400];
(%o100) [gr2d(explicit,points)]

9. zadatak (prekid druge vrste)

(%i101) f(x):=sin(1/x);
(%o101) f(x):=sin(1/x)

oba jednostrana limesa osciliraju u okolini nule

(%i102) limit(f(x),x,0,minus);
(%o102) ind
(%i103) limit(f(x),x,0,plus);
(%o103) ind
(%i104) wxdraw2d(grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,3],ytics=[-1.2,0.2,1.2],
line_width=3,color=blue,explicit(f(x),x,-3,3),
yrange=[-1.2,1.2]),wxplot_size=[600,400];
(%t104)
(Graphics)
(%o104)
(%i105) draw2d(terminal=wxt,grid=true,xaxis=true,xaxis_width=1,xaxis_type=solid,xlabel="x",
yaxis=true,yaxis_width=1,yaxis_type=solid,ylabel="y",xtics=[-3,1,3],ytics=[-1.2,0.2,1.2],
line_width=3,color=blue,explicit(f(x),x,-3,3),
yrange=[-1.2,1.2]),wxplot_size=[600,400];
(%o105) [gr2d(explicit)]
Created with wxMaxima.