Final answer:
For the curve y = 1/x, as x approaches infinity, the curvature diminishes and tends to zero, indicating that the curve flattens out and approaches a horizontal asymptote at y = 0.
Step-by-step explanation:
To understand what happens to the curvature as x approaches infinity for the curve y = 1/x, we refer to the concept of asymptotes in functions. As x becomes very large (approaches infinity), the value of y gets closer and closer to zero, but never actually reaches it. This means that the curve y = 1/x gets closer to the x-axis without touching it, which implies the curve has a horizontal asymptote at y = 0.
The curvature of a curve tells us how sharply the curve bends at a given point. For y = 1/x, as x increases, the amount of bend in the curve decreases because the change in y for each additional unit of x decreases. Therefore, the curvature diminishes as x approaches infinity, and tends to zero. Hence, the curve flattens out as it extends infinitely to the right along the x-axis.