How do you graph #f(x)=(2x^3+x^28x4)/(x^23x+2)# using holes, vertical and horizontal asymptotes, x and y intercepts?
see below
graph{(2x^3+x^28x4)/(x^23x+2) [200, 200, 150, 150]}
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To graph the function f(x) = (2x^3 + x^2  8x  4)/(x^2  3x + 2), follow these steps:

Find the xintercepts by setting f(x) = 0 and solving for x. In this case, the numerator can be factored as (2x + 1)(x^2  4x  4), and the denominator can be factored as (x  1)(x  2). Therefore, the xintercepts are x = 1/2, x = 1, and x = 2.

Find the yintercept by evaluating f(0). Substitute x = 0 into the function, giving f(0) = 4/2 = 2. So the yintercept is (0, 2).

Determine any vertical asymptotes by finding the values of x that make the denominator equal to zero. In this case, the denominator factors as (x  1)(x  2), so the vertical asymptotes are x = 1 and x = 2.

Determine any holes in the graph by canceling out common factors between the numerator and denominator. In this case, (2x + 1) cancels out with (x  1), leaving a hole at x = 1/2.

Determine any horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote.

Plot the xintercepts, yintercept, vertical asymptotes, and the hole on the graph.

Determine the behavior of the graph as x approaches positive and negative infinity. Since there is no horizontal asymptote, the graph will approach infinity or negative infinity as x approaches positive or negative infinity, respectively.

Sketch the graph, connecting the points and following the behavior described above.
This completes the process of graphing the function f(x) = (2x^3 + x^2  8x  4)/(x^2  3x + 2).
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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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