How do you graph #f(x)=(x^2x2)/(x^22x+1)# using holes, vertical and horizontal asymptotes, x and y intercepts?
graph{(x^2x2)/(x^22x+1) [10, 10, 4.64, 5.36]}
holes are whatever factors cancels out from the numerator and denominator. What ever factors are left in the denominator if it can equal 0 whatever x equaled for it to be 0 would be the V.A. There is no H.A because both exponents of the numerator and denominator are the same and their coefficients are 1.
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To graph the function f(x) = (x^2  x  2)/(x^2  2x + 1), we can analyze its properties:

Holes: To find any holes in the graph, we need to identify values of x that make the numerator and denominator equal to zero. In this case, the numerator factors as (x  2)(x + 1), and the denominator factors as (x  1)(x  1). Therefore, the function has a hole at x = 2.

Vertical Asymptotes: Vertical asymptotes occur when the denominator of a rational function equals zero, but the numerator does not. In this case, the denominator (x  1)(x  1) equals zero at x = 1. Thus, the function has a vertical asymptote at x = 1.

Horizontal Asymptotes: To determine the horizontal asymptote(s), we compare the degrees of the numerator and denominator. In this case, both have a degree of 2. Therefore, we divide the leading coefficients of the numerator and denominator, which gives us 1/1 = 1. Hence, the function has a horizontal asymptote at y = 1.

xintercepts: To find the xintercepts, we set the numerator equal to zero and solve for x. In this case, (x  2)(x + 1) = 0, which gives us x = 2 and x = 1 as the xintercepts.

yintercept: To find the yintercept, we substitute x = 0 into the function. Thus, f(0) = (2)/1 = 2, giving us the yintercept at (0, 2).
By considering these properties, we can plot the graph of f(x) = (x^2  x  2)/(x^2  2x + 1) accordingly.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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