Question

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

Answer 1

$f(x)=\frac{3x}{4-x}$

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:

$\lim_{x \to \infty} (\frac{3x}{4-x})=-3 \\ \lim_{x \to -\infty} (\frac{3x}{4-x})=-3$

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:

$\lim_{x \to \infty} (\frac{3x}{4-x}) =\frac{\infty}{-\infty}\\=^L \lim_{x \to \infty} (\frac{3}{-1}) = -3\\\\ \lim_{x \to- \infty} (\frac{3x}{4-x}) =\frac{-\infty}{\infty}\\=^L \lim_{x \to- \infty} (\frac{-3}{1}) = -3$

Graphed by: Desmos