Question

Which statement describes the behavior of the function f(x)=3x/4-x?

a. The graph approaches –3 as x approaches infinity.
b. The graph approaches 0 as x approaches infinity.
c. The graph approaches 3 as x approaches infinity.
d. The graph approaches 4 as x approaches infinity.

Answer 1

a. The graph approaches –3 as x approaches infinity.


`f(x) = (3x)/(4-x) +3 -3 = (3x+12-3x)/(4-x) -3 = (12)/(4-x) -3`

Answer 2

The graph approaches –3 as x approaches infinity. Option a is correct.

Explanation:

The given function is

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

We have to find value of function as x approaches infinity. Take limit both sides as x approaches to infinity.

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

Taking x common from the denominator.

`\lim_(x\rightarrow \infty)f(x)=\lim_(x\rightarrow \infty)(3x)/(x((4)/(x)-1))`

Cancel out common factor x.

`\lim_(x\rightarrow \infty)f(x)=\lim_(x\rightarrow \infty)(3)/((4)/(x)-1)`

Apply limits.

`\lim_(x\rightarrow \infty)f(x)=(3)/((4)/(\infty)-1)`

`\lim_(x\rightarrow \infty)f(x)=(3)/(0-1)`

`\lim_(x\rightarrow \infty)f(x)=-3`

Therefore the graph approaches –3 as x approaches infinity.