Ilustração Gráfica dos Limites

Questão 90

Item a)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) sin(3*x)/(2*x), col="red") +
  geom_hline(yintercept = 1.5,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.6, y = 1.4, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(sen~3*x,2*x)==frac(3,2)")

Item b)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) sin(2*x)/(x), col="red") +
  geom_hline(yintercept = 2,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.7, y = 1.9, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(sen~2*x,x)==2")

Item e)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) tan(2*x)/(3*x), col="red") +
  geom_hline(yintercept = 2/3,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.35, y = 38, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(tg~2*x,3*x)==frac(2,3)")

Item g)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) (1-cos(x))/x, col="red") +
  geom_hline(yintercept = 0,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.7, y = 0.15, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(1-cos~x,x)==0")

Item h)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) (1-1/cos(x))/x^2, col="red") +
  geom_hline(yintercept = -0.5,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.7, y = 0, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(1-sec~x,x^2)==-frac(1,2)")

Item i)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) (tan(x)+sin(x))/x, col="red") +
  geom_hline(yintercept = 2,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.4, y = 2.3, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(tg~x+sen~x,x)==2")

Item j)

ggplot()+
  xlim(-1,1)+
  geom_function(fun = function(x) (1 - cos(x))/(x*sin(x)), col="red") +
  geom_hline(yintercept = 1/2,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.4, y = 0.55, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~frac(1-cos~x,x~sen~x)==frac(1,2)")

Questão 92

Item d)

ggplot()+
  xlim(pi/4-0.2,pi/4+0.2)+
  geom_function(fun = function(x) (sin(x)-cos(x))/(1-tan(x)), col="red")+
  geom_hline(yintercept = -sqrt(2)/2, col="blue")+
  geom_vline(xintercept = pi/4, col="green")+
  annotate("text", x=0.9, y=-0.50, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~~pi/4]~frac(sen~x-cos~x,1-tg~x)==-frac(sqrt(2),2)")

Item e)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (tan(x)-sin(x))/(sin(x))^2, col="red")+
  geom_hline(yintercept = 0, col="blue")+
  geom_vline(xintercept = 0, col="green")+
  annotate("text", x=0.2, y=0.5, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(tg~x-sen~x,sen^2~x)==0")

Item f)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (sin(3*x)-sin(2*x))/sin(x), col="red")+
  geom_hline(yintercept = 1, col="blue")+
  geom_vline(xintercept = 0, col="green")+
  annotate("text", x=-0.090, y=0.5, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(sen~3*x-sen~2*x,sen~x)==1")

Item g)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (cos(3*x)-cos(2*x))/x^2, col="red")+
  geom_hline(yintercept = -5/2, col="blue")+
  geom_vline(xintercept = 0, col="green")+
  annotate("text", x=0.25, y=-1.5, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(cos~3*x-cos~2*x,x^2)==frac(5,2)")

Item j)

ggplot()+
  xlim(pi-0.5,pi+0.5)+
  geom_function(fun = function(x) (1-sin(x/2))/(pi-x) , col="red")+
  geom_hline(yintercept = 0, col="blue")+
  geom_vline(xintercept = pi, col="green")+
  annotate("text", x=3.5, y= 0.04, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~pi]~frac(1-sen~frac(x,2),pi-x)==0")

Item k)

ggplot()+
  xlim(pi/3-0.5,pi/3+0.5)+
  geom_function(fun = function(x) (1-2*cos(x))/(pi-3*x) , col="red")+
  geom_hline(yintercept = -sqrt(3)/3, col="blue")+
  geom_vline(xintercept = pi, col="green")+
  annotate("text", x= 1.30, y= -0.5, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~pi/3]~frac(1-2*cos~x,pi-3*x)==-frac(sqrt(3),3)")

Item l)

ggplot()+
  xlim(-0.5,1.5)+
  geom_function(fun = function(x) (1-x^2)/sin(pi*x), col = "red")+
  geom_hline(yintercept = 2/pi, col="blue")+
  geom_vline(xintercept = 1, col="green")+
  annotate("text", x= 1.30, y= 8, parse = TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~1]~frac(1-x^2,sen~pi*x)==frac(2,pi)")

Item m)

ggplot()+
  xlim(pi/4-0.4,pi/4+0.4)+
  geom_function(fun = function(x) cos(2*x)/(cos(x)-sin(x)), col = "red")+
  geom_vline(xintercept = pi/4, col = "green")+
  geom_hline(yintercept = sqrt(2), col = "blue")+
  annotate("text", x= 0.98, y = 1.5, parse= TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~pi/4]~frac(cos~2*x,cos~x-sen~x)==sqrt(2)")

Item n)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (1-cos(x)^3)/(sin(x)^2), col = "red")+
  geom_vline(xintercept = 0, col = "green")+
  geom_hline(yintercept = 3/2, col = "blue")+
  annotate("text", x= 0.38, y = 1.485, parse= TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(1-cos^3~x,sen^2~x)==frac(3,2)")

Item q)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (x-sin(2*x))/(x+sin(3*x)), col = "red")+
  geom_vline(xintercept = 0, col = "green")+
  geom_hline(yintercept = -1/4, col = "blue")+
  annotate("text", x= 0.25, y = -0.22, parse= TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(x-sen~2*x,x+sen~3*x)==-frac(1,4)")

Item r)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (1-cos(x))/x^2, col = "red")+
  geom_vline(xintercept = 0, col = "green")+
  geom_hline(yintercept = 1/2, col = "blue")+
  annotate("text", x= 0.25, y = 0.505, parse= TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(1-cos~x,x^2)==frac(1,2)")

Item s)

ggplot()+
  xlim(0.5,1.5)+
  geom_function(fun = function(x) cos(pi*x/2)/(1-x), col = "red")+
  geom_vline(xintercept = 1, col = "green")+
  geom_hline(yintercept = pi/2, col = "blue")+
  annotate("text", x= 1.35, y = 1.55, parse= TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~1]~frac(cos(frac(pi*x,2)), 1-x)==frac(pi,2)")

Item t)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) (sqrt(1+sin(x)) - sqrt(1-sin(x)))/x, col = "red")+
  geom_vline(xintercept = 0, col = "green")+
  geom_hline(yintercept = 1, col = "blue")+
  annotate("text", x= 0.25, y = 1.002, parse= TRUE, size = 4,
           label = "'Limite: ' * lim[x%->%~0]~frac(sqrt(1+sen(x)) - sqrt(1-sen(x)),x)==1")

Questão 93

Item a)

ggplot()+
  xlim(-0.5,0.5)+
  geom_function(fun = function(x) x*sin(1/x), col="red") +
  geom_hline(yintercept = 0,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.2, y = 0.55, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%0]~x%.%sen~frac(1,x)==~0")

Item b)

ggplot()+
  xlim(-3,8)+
  geom_function(fun = function(x) x*sin(1/x), col="red") +
  geom_hline(yintercept = 1,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 3.2, y = 0.90, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%~infinity]~x%.%sen~frac(1,x)==~1")

Item c)

ggplot()+
  xlim(0.5, 1.5)+
  geom_function(fun = function(x) ((1 - x)*tan((pi*x)/2)), col="red") +
  geom_hline(yintercept = 2/pi,col="blue")+
  geom_vline(xintercept = 1,col="green")+
  annotate("text", x =1.2, y = 0.75, parse = TRUE, size = 4,
           label = "'Limite:  ' * lim[x%->%~1]~(1-x)%.%tan~frac(pi*x,2)==~frac(2,pi)")

Item d)

ggplot()+
  xlim(-0.5, 0.5)+
  geom_function(fun = function(x) (1/tan(2*x))*(1/tan(pi/2 - x)), col="red") +
  geom_hline(yintercept = 1/2,col="blue")+
  geom_vline(xintercept = 0,col="green")+
  annotate("text", x = 0.05, y = 0.39, parse = TRUE, size = 4,
          label = "'Limite:  ' * lim[x%->%0]~cotg(2%.%x)%.%cotg~(frac(pi,2)- x)==~frac(1,2)")