Question

Which statement describes the behavior of the function f (x) = startfraction 2 x over 1 minus x squared endfraction?

Answer 1

The statement that describes the behavior of the function f(x) = 2x/(1-x²) is given by: Option B: The graph approaches 0 as x approaches infinity

How to find the value of the function as x approaches infinity (+ve or -ve)?

If limits exist, we can take limits of the function, where x tends to -∞ or ∞, and that limiting value will be the value the function will approach.

For the considered case, the function is:

$f(x) = \dfrac{2x}{1-x^2}$

The missing options are:

1. The graph approaches –2 as x approaches infinity.
2. The graph approaches 0 as x approaches infinity.
3. The graph approaches 1 as x approaches infinity.
4. The graph approaches 2 as x approaches infinity.

So we need to find the limit of the function as x approaches infinity.

$lim_{x\rightarrow \infty}f(x) = lim_{x\rightarrow \infty}\dfrac{2x}{1-x^2} = lim_{x\rightarrow \infty}\dfrac{2x/x^2}{1/x^2 - 1} = lim_{x\rightarrow \infty}\dfrac{2/x}{1/x^2 - 1}\\\\lim_{x\rightarrow \infty}f(x) = \dfrac{lim_{x\rightarrow \infty}(2/x^2)}{lim_{x\rightarrow \infty}(1/x^2 - 1)} = \dfrac{0}{0-1} = \dfrac{0}{-1} = 0$

Since the value of the function approaches 0 as x approaches infinity, so as its graph will do.

Thus, the statement that describes the behavior of the function f(x) = 2x/(1-x²) is given by: Option B: The graph approaches 0 as x approaches infinity

Learn more about limits of a function here:

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