To analyze the behavior of the function [tex]\( f(x) = \frac{3x}{4-x} \)[/tex] as [tex]\( x \)[/tex] approaches infinity, let's break down the expression step-by-step.
1. Identify Dominant Terms:
When [tex]\( x \)[/tex] becomes very large, the constant term 4 in the denominator becomes relatively insignificant compared to [tex]\( x \)[/tex]. Thus, it is useful to focus on the terms involving [tex]\( x \)[/tex] because they will dominate the behavior of the function.
2. Simplify the Expression for Large [tex]\( x \)[/tex]:
For large values of [tex]\( x \)[/tex]:
[tex]\[
f(x) = \frac{3x}{4 - x}
\][/tex]
Since [tex]\( x \)[/tex] is very large, [tex]\( 4 - x \)[/tex] is approximately equal to [tex]\( -x \)[/tex]:
[tex]\[
f(x) \approx \frac{3x}{-x}
\][/tex]
3. Reduce the Simplified Expression:
Dividing the numerator and the denominator by [tex]\( x \)[/tex]:
[tex]\[
f(x) \approx \frac{3}{-1} = -3
\][/tex]
4. Conclusion:
As [tex]\( x \)[/tex] approaches infinity, the function [tex]\( f(x) = \frac{3x}{4 - x} \)[/tex] approaches the value [tex]\( -3 \)[/tex].
Thus, the correct statement is:
The graph approaches [tex]\(-3\)[/tex] as [tex]\( x \)[/tex] approaches infinity.
