I will try to write a proof that proves the limit as x approaches 2 of the function f(x)=(-4+x^2)/(2-3x+x^2) equals 0. I thought this limit seemed wrong. The limit should be 4, not 0.
What I do not understand as of right now is that it seems that the limit of (-4+x^2)/(2-3x+x^2) as x approaches 2 would be 4 because that is where the hole is in the function.
What is easy about this is that I really like the example given to us beforehand. What will be hard is trying to figure out exactly what the formulas are for this problem to work out correctly and give me the answer of lim x→2 ((-4+x^2)/(2-3x+x^2))=0. The example we were given was that the limit as x approaches -2 of the function (x+2)/(x^2+3x+2) equals -1. The reason or this even though there is a hole at x=-2 is because there is a constant, C such that f(x)-(-1)<C|x-(-2)|. I am assuming that the -1 is for the limit and the -2 is for the x that the limit is approaching. Additionally, there is a seemingly random fraction, ½, that is set as greater than or equal to the absolute value of x+2. I tried plugging in different values of x to try to obtain ½, and I believe I got x=-3 for the example problem, but this does not help me because it seems this integer comes from nowhere.
I have done some researching and some speculating and I have come to the conclusion that we use ½ to set a boundary for our x to be in. We do not want to use one in this case or else the limit could be approaching -3 or -1. So to restrict our x value, we set the change in x to be ½ and see what x value the limit approaches.
A much simpler way of proving that the limit as x approaches 2 of the function ((-4+x^2)/(2-3x+x^2)) is to take the derivative of the numerator and the denominator and then plug in 2 for x. Taking the derivative of the numerator gives us 2x and taking the derivative of the denominator gives us 2x-3. Then we get 2(2)/(2(2)-3) which equals 4/1 which equals 4.
The limit as x approaches 8 of the function (x+4)/(x^2 – 10x +10) is equal to -2. We can prove this simply by plugging in 8 for x in the function. (8+4)/(8^2 – 10(8) +10) = 12/(64-80+10) =
12/-6 = -2.
What was hard about trying to write this proof is that I have never thought about finding a limit in this way. We were taught to find where the hole is in a function and then we find the limit that way. At least the second function was easier to solve for the limit because I just needed to plug in the x value.