How do you graph and find the discontinuities of #f(x) = (x  1) / (x^2  x  6)#?
This function is discontinuous at
Such function is discontinuous at the points, which are the zeros of denominator, but are not zeroes of the numerator.
graph{(x1)/(x^2x6) [10, 10, 5, 5]}
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To graph and find the discontinuities of f(x) = (x  1) / (x^2  x  6), follow these steps:

Factor the denominator: x^2  x  6 = (x  3)(x + 2).

Set the denominator equal to zero and solve for x: (x  3)(x + 2) = 0. This gives x = 3 and x = 2 as the values that make the denominator zero.

These values, x = 3 and x = 2, are the potential discontinuities of the function.

Determine the behavior of the function around these values. To do this, evaluate the function for values of x slightly greater and slightly smaller than the potential discontinuities.

Create a number line and mark the potential discontinuities on it.

Test a value slightly smaller than 2, such as 3, in the function. Evaluate f(3) = (3  1) / ((3)^2  (3)  6) = 4 / 18 = 2/9.

Test a value slightly greater than 2, such as 1, in the function. Evaluate f(1) = (1  1) / ((1)^2  (1)  6) = 2 / 8 = 1/4.

Test a value slightly smaller than 3, such as 2, in the function. Evaluate f(2) = (2  1) / (2^2  2  6) = 1 / 0, which is undefined.

Test a value slightly greater than 3, such as 4, in the function. Evaluate f(4) = (4  1) / (4^2  4  6) = 3 / 6 = 1/2.

Based on the evaluations, the function has a vertical asymptote at x = 3 and a hole at x = 2.

Plot the points (2, 1/4) and (3, undefined) on the graph.

Draw the graph, approaching the vertical asymptote at x = 3 and having a hole at x = 2.
This is how you graph and find the discontinuities of f(x) = (x  1) / (x^2  x  6).
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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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