#Thomas Scalese
#Learning To Look At Limits

# Question 1. Find the limit of the function F(x) = x^2 + 3*x + 2, as x approaches 3 from the left and right side of the graph. Then determine what the limit of f(x) is when x approaches 3 from both sides. Then plot the function, and use the correct paramters, to visualize the limit as f(x) approaches 3.

f(x) = x^2 + 3*x + 2
N(f(2.9))
N(f(2.99))
N(f(2.999))
#as x approaches 3 from the left side, f(x) approaches 20

N(f(3.1))
N(f(3.01))
N(f(3.001))
#as x approaches 3 from the right side, f(x) approaches 20

#Becasue as x approaches 3 from the left side, f(x) approaches 20, and as x approaches 3 from the right side, f(x) also approaches 20, we can determine that the limit of f(x) as x approaches 3 from both sides is 20. We can also solve this mathematicaly by plugging in 3 for x in f(x)
N(f(3))

plot(x^2 + 3*x +2, 0,4) + plot(20, 0,4, color="red")

# Question 2. Find the limit of the funcation, f(x) = (x-4)/(x-2), as x approaches 2 from the left and right side. Then determine what the limit of f(x) as x approaches 2 from both sides is. If the limit does not exist please state what type of discontinuity is.

f(x) = (x-4)/(x-2)
N(f(1.9))
N(f(1.99))
N(f(1.999))
N(f(1.9999))
N(f(1.99999))
N(f(1.999999))
#as x approaches 2 from the left side, f(x) approaches positive infinity

N(f(2.1))
N(f(2.01))
N(f(2.001))
N(f(2.0001))
N(f(2.00001))
N(f(2.000001))
#as x approaches 2 from the right side, f(x) approaches negative infinity

plot((x-4)/(x-2), 0,4)
#Because the limits of f(x) as x approaches 2 from the left and right side are postive infinity and negative infinity respectively you can say that the limit of f(x), when x approaches 2, does not exist. You can also see this by plotting the graph, the graph will never touch x=2. Becasue of this the funcation has an infinite disontinuity.

#Question 3. Find the limit of the funcation, f(x) = (x)/(abs(x)), as x approaches 0 from the left and right sides of the graph. Then with that information determine what the limit of f(x) when x approaches 0 from both sides is. If the limit does not exist please determine what type of disontinuity it is.

f(x) = (x)/(abs(x))
N(f(-0.1))
N(f(-0.01))
N(f(-0.001))
#as x approaches 0 from the left side, f(x) approaches -1

N(f(0.1))
N(f(0.01))
N(f(0.001))
#as x approaches 0 from the right side, f(x) approaches 1

plot((x)/(abs(x)))
#with this information you can determine that the limit of f(x) as x appraoches 0 from both sides, does not exist. You can also see this by graphing f(x). On the graph you can see that f(x) will never touch as one sigle point at x=0. This is considered a jump discontinuity.

#Question 4. Find the limit of the funcation, f(x) = (x^2 - 2*x -3)/(x+1), as x approaches -1 from the left and right sides of the graph. Then determine what f(x)=-1 is. Determine if there is a removable discontinuity.

f(x) = (x^2 - 2*x -3)/(x+1)
N(f(-0.9))
N(f(-0.99))
N(f(-0.999))
#as x apporaches -1 from the left side, f(x) approaches -4

N(f(-1.1))
N(f(-1.01))
N(f(-1.001))
#as x approaches -1 from the right side, f(x) appraoches -4

#With this function f(x) you are unable to plug in x=-1. This is because the denominator can never equal 0. When you factor the numerator you get, (x-3)(x+1)/(x+1). You can cancel out the (x+1), so therefor you are left with (x-3).

#Becasue you were able to cancel somthing out these means that there is a hole in the graph, which means there is a removable dicontinuity at x=-1. In order to find where the removable discontinuty is you should plug x=-1 into the new funcation.
g(x) = (x-3)
N(g(-1))
#This means the point of removeable diconsinuity is at (-1,-4)

#What is also interesting to point out is that sage graphs does not show holes in the graph.
plot((x^2 - 2*x -3)/(x+1), -2,0)

#Question 5. In the iconic and legendary movie, Mean Girls, the main character Cady Heron, played by Lindsay Lohan, is on the Mathletes. During a Mathelete competition Cady is tasked with finding the limit as x approaches 0, of f(x) = (ln(1-x)-sinh(x))/(1-cosh(x)^2). She comes to the conclusion that "The Limit Does Not Exist". Please explain numerically, and then graphicly, why the limit does not exist.

f(x) = (ln(1-x)-sinh(x))/(1-cosh(x)^2)
N(f(-0.1))
N(f(-0.01))
N(f(-0.001))
#as x approaches 0 from the left side, f(x) approaches negative infinity

N(f(0.1))
N(f(0.01))
N(f(0.001))
#as x approaches 0 from the right side, f(x) approaches positive infinity

#Because the limits of f(x) as x approaches 0 from the left and right side are negative infinity and positive infinity respectively you can say that the limit of f(x), as x approaches 0, does not exist. This is also an example of infinite disontinuity.

plot((ln(1-x)-sinh(x))/(1-cosh(x)^2), -0.01,0.01)
#When you plot the graph you can see as x apporoaches 0 from the left side, f(x) will never touch x=0 and contunitue to negative infinity. Also when x apporoaches 0 from the right side, f(x) will never touch x=0 and continue to positive infinity. Becasue of this the limit of f(x) as x approaches 0, does not exist.