Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Accepted Answer

First, let's understand the conditions given for the function f(x). The condition lim x → ±∞ f(x) = 0 suggests that as x approaches positive or negative infinity, the function approaches zero. This is characteristic of functions that have horizontal asymptotes at y = 0. Next, consider the condition lim x → 2⁻ f(x) = ∞ and lim x → 2⁺ f(x) = ∞. These conditions indicate that as x approaches 2 from the left and right, the function approaches infinity. This suggests a vertical asymptote at x = 2. A function that satisfies these conditions could be a rational function with a vertical asymptote at x = 2 and horizontal asymptotes at y = 0. One example is f(x) = $$1/(x-2)^2. $$This function has a vertical asymptote at x = 2 and approaches zero as x approaches ±∞. To sketch the graph of f(x) = $$1/(x-2)^2$$, note that the graph will have a vertical asymptote at x = 2, meaning the function will shoot up to infinity as x approaches 2 from either side. The graph will also approach the x-axis (y = 0) as x moves towards positive or negative infinity. Finally, verify the behavior of the function at the asymptotes. As x approaches 2 from the left (x → 2⁻), f(x) = $$1/(x-2)^2$$ tends to infinity, and similarly, as x approaches 2 from the right (x → 2⁺), f(x) also tends to infinity. As x approaches ±∞, f(x) approaches 0, confirming the horizontal asymptote at y = 0.

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Key Concepts

Limits at Infinity

Vertical Asymptotes

Piecewise Functions