Vertical Asymptote at x = 2, hole when x = 1
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Vertical asymptote at x = 2 means that the function approaches infinity as x gets closer and closer to 2.
A hole at x = 1 means that the function approaches a specific value as x gets closer and closer to 1, but there is a gap or hole in the graph at that point.
One example of a function that has these characteristics is:
f(x) = (x-1)/(x-1)(x-2)
When x = 2, the denominator becomes 0, so the function becomes undefined and approaches infinity. Therefore, there is a vertical asymptote at x = 2.
When x = 1, the numerator and denominator both become 0, so there is a hole in the graph at that point. However, if we simplify the function by canceling out the common factors, we get:
f(x) = 1/(x-2)
This function has a vertical asymptote at x = 2, and as x approaches 1, the function approaches -1. Therefore, the function has a hole at x = 1 and a vertical asymptote at x = 2.
More Answers:
Understanding Horizontal Asymptotes and Points of Discontinuity in Functions.Exploring a Mathematical Function with Horizontal and Vertical Asymptotes: Understanding f(x) = k/x
Creating a Function with a Hole at x = -1 and Vertical Asymptote at x = 1