How do you graph #f(x)=(2x1)/(x+3)# using holes, vertical and horizontal asymptotes, x and y intercepts?
Below
After plotting the intercepts and the asymptotes, you should get something like this graph{(2x1)/(x+3) [10, 10, 5, 5]}
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To graph the function f(x) = (2x1)/(x+3), follow these steps:

Determine the vertical asymptote by finding the values of x that make the denominator (x+3) equal to zero. In this case, x = 3 is the vertical asymptote.

Find any holes in the graph by canceling out common factors between the numerator and denominator. In this case, there are no common factors to cancel, so there are no holes.

Calculate the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (1), the horizontal asymptote is y = 0.

Find the xintercept by setting f(x) equal to zero and solving for x. In this case, there is no xintercept.

Find the yintercept by evaluating f(0). Substitute x = 0 into the function: f(0) = (2(0)1)/(0+3) = 1/3. Therefore, the yintercept is (0, 1/3).

Plot the vertical asymptote at x = 3, the horizontal asymptote at y = 0, and the yintercept at (0, 1/3).

Choose additional xvalues and calculate the corresponding yvalues to plot more points on the graph. For example, you can choose x = 4, 2, 1, and 2.

Connect the plotted points smoothly, avoiding the vertical asymptote.
This completes the graph of f(x) = (2x1)/(x+3).
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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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