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

#x = 1# the vertical asymptote of#f(x)# #y = 0# the horizontal asymptote of#f(x)# #(0, 2)# the#y# -intercept#f(x)# has no#x# -intercept

graph{2/(1-x) [-10, 10, -5, 5]}

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

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

Horizontal asymptote: To determine the horizontal asymptote, we need to compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 0 (constant) and the degree of the denominator is 1. Since the degree of the denominator is greater, the horizontal asymptote is y = 0.

Hole: A hole occurs when both the numerator and denominator have a common factor that can be canceled out. In this case, there is no common factor, so there is no hole.

X-intercept: To find the x-intercept, we set y (or f(x)) equal to zero and solve for x. In this case, we have 2/(x-1) = 0. Since the numerator is never zero, there are no x-intercepts.

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

To summarize:

- Vertical asymptote: x = 1
- Horizontal asymptote: y = 0
- Hole: None
- X-intercept: None
- Y-intercept: (0, -2)

Using this information, you can plot the graph of f(x) = 2/(x-1) accordingly.

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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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