# How do you graph #f(x)=(2x)/(3x-1)# using holes, vertical and horizontal asymptotes, x and y intercepts?

So here:

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To graph the function f(x) = (2x)/(3x-1), we can start by identifying the vertical and horizontal asymptotes, x and y intercepts, and any holes in the graph.

Vertical asymptote: The vertical asymptote occurs when the denominator of the function becomes zero. In this case, the denominator 3x-1 becomes zero when x = 1/3. Therefore, the vertical asymptote is x = 1/3.

Horizontal asymptote: To find the horizontal asymptote, we compare the degrees of the numerator and denominator. In this case, both the numerator and denominator have a degree of 1. Therefore, the horizontal asymptote is y = 2/3.

X-intercept: To find the x-intercept, we set y (or f(x)) equal to zero and solve for x. In this case, setting (2x)/(3x-1) = 0, we find that x = 0. Therefore, the x-intercept is (0, 0).

Y-intercept: To find the y-intercept, we set x equal to zero and evaluate the function. In this case, setting x = 0, we find that f(0) = 0. Therefore, the y-intercept is (0, 0).

Holes: To check for any holes in the graph, we simplify the function and see if any common factors cancel out. In this case, there are no common factors that cancel out, so there are no holes in the graph.

Using this information, we can plot the vertical asymptote at x = 1/3, the horizontal asymptote at y = 2/3, the x-intercept at (0, 0), and the y-intercept at (0, 0).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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