To understand the behavior of the function [tex]\( f(x) = \frac{3x}{4 - x} \)[/tex] as [tex]\( x \)[/tex] approaches infinity, let's perform a detailed step-by-step analysis:
1. Function Analysis:
- We start with the function [tex]\( f(x) = \frac{3x}{4 - x} \)[/tex].
2. Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large (approaches infinity), the term [tex]\( -x \)[/tex] in the denominator will dominate over the constant 4. In other words, [tex]\( 4 - x \approx -x \)[/tex] when [tex]\( x \)[/tex] is very large.
3. Approximation:
- Thus, the function can be approximated by replacing the denominator with [tex]\( -x \)[/tex] for large values of [tex]\( x \)[/tex]:
[tex]\[
f(x) \approx \frac{3x}{-x}
\][/tex]
4. Simplification:
- We simplify the fraction:
[tex]\[
\frac{3x}{-x} = -3
\][/tex]
- This means that as [tex]\( x \)[/tex] approaches infinity, [tex]\( f(x) \)[/tex] approaches [tex]\(-3\)[/tex].
5. Conclusion:
- The function [tex]\( f(x) = \frac{3x}{4 - x} \)[/tex] approaches [tex]\(-3\)[/tex] as [tex]\( x \to \infty \)[/tex].
Thus, the correct statement is:
The graph approaches -3 as [tex]\( x \)[/tex] approaches infinity.
