Question

Which statement describes the behavior of the function f (x) = StartFraction 3 x Over 4 minus x EndFraction?

Answer 1

`f(x)=(3x)/(4-x)`

First Method: Using Graph

Finding the limits as x approaches infinity and negative infinity is one way to solve this problem. As x reaches infinity, simply follow the graph line (colored purple) to the far right to find its limit. If we trace it, we can see that the Y value never exceeds -3 (orange), indicating that the limit is equal to -3 as x approaches infinity. You'd do the same with negative infinity, the limit is -3. We may now say the following:

`\lim_(x \to \infty) ((3x)/(4-x))=-3 \\ \lim_(x \to -\infty) ((3x)/(4-x))=-3`

And that's the answer to your question.

Second Method: Using Mathematics

I'm not sure if this solution is suitable for your stage, but you can solve this problem using L'Hopital's rule:

`\lim_(x \to \infty) ((3x)/(4-x)) =(\infty)/(-\infty)\\=^L \lim_(x \to \infty) ((3)/(-1)) = -3\\\\ \lim_(x \to- \infty) ((3x)/(4-x)) =(-\infty)/(\infty)\\=^L \lim_(x \to- \infty) ((-3)/(1)) = -3`

Graphed by: Desmos

Answer 2

A: the graph approaches-3 as x approaches infinity

Explanation: