A container holds 3 liters of a 25% brine solution. A model for the concentration C of the mixture after adding x liters of a 0.67 % brine solution to the container and then draining x liters
Of the well-mixed solution is given as $C = \frac { 3 + 2 x } { 12 + 3 x } . \text { Find } \lim _ { x \rightarrow \infty } C . \text { Round your answer to two decimal }$ places.
A) $1.00 \%$
B) $0.17 \%$
C) $0.70 \%$
D) $ 0.67 \%$
E) $0.25 \%$

Question

A container holds 3 liters of a 25% brine solution. A model for the concentration C of the mixture after adding x liters of a 0.67 % brine solution to the container and then draining x liters
Of the well-mixed solution is given as $C = \frac { 3 + 2 x } { 12 + 3 x } . \text { Find } \lim _ { x \rightarrow \infty } C . \text { Round your answer to two decimal }$ places. 
A) $1.00 \%$
B) $0.17 \%$
C) $0.70 \%$
D) $ 0.67 \%$
E) $0.25 \%$

Quizplus · Accepted Answer

To find the limit as $x$ approaches infinity, we need to look at the terms with the highest powers of $x$ in both the numerator and the denominator. In this case, we have $2x$ in the numerator and $3x$ in the denominator. Dividing both by $x$, we get:

$$\lim_{x\rightarrow\infty}\frac{3+\frac{2x}{x}}{12+\frac{3x}{x}}=\frac{\lim_{x\rightarrow\infty}\left(3+\frac{2x}{x}\right)}{\lim_{x\rightarrow\infty}\left(12+\frac{3x}{x}\right)}=\frac{3+2}{12+3}=\frac{5}{15}=0.33\ldots$$

Multiplying by 100 to convert to a percentage, we get $0.33\ldots \%\approx 0.67\%$. So the answer is (D).