How do you graph and find the discontinuities of #(x^2)/(x-1)#?

Question

Julia Adams · Accepted Answer

The function is discontinuous at #x=1# We notice that as #x->1# we have that #x^2/(x-1)->oo#. So at #x=1# there is a discontinuity. The graph is as follows graph{x^2/(x-1) [-20, 20, -10, 10]}

Nathan Axel · Answer

undefined To graph and find the discontinuities of the function (x^2)/(x-1), follow these steps:

1. Determine the vertical asymptotes: Set the denominator equal to zero and solve for x. In this case, (x-1) = 0, so x = 1. The vertical asymptote is x = 1.

2. Determine the horizontal asymptote: Divide the leading terms of the numerator and denominator. In this case, the leading terms are both x^2, so the horizontal asymptote is y = 1.

3. Find the x-intercept: Set the numerator equal to zero and solve for x. In this case, x^2 = 0, so x = 0. The x-intercept is (0, 0).

4. Find the y-intercept: Substitute x = 0 into the function. In this case, (0^2)/(0-1) = 0. The y-intercept is (0, 0).

5. Plot the points and draw the graph: Plot the vertical asymptote, horizontal asymptote, x-intercept, and y-intercept on a coordinate plane. Then, draw a smooth curve that approaches the asymptotes and passes through the intercepts.

6. Determine the discontinuities: The function has a discontinuity at x = 1, which is the vertical asymptote. There are no other discontinuities in this case.

Remember to label the axes and any other relevant points on the graph.