Let $f(x)=\frac{6 x^{3}}{3 x+2}$ . $\newline$ Find $\lim _{x \rightarrow \infty} f(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $2$ $\newline$ (B) $3$ $\newline$ (C) $0$ $\newline$ (D) The limit is unbounded

Full solution

Q. Let

f(x)=\frac{6 x^{3}}{3 x+2}

\newline

Find

\lim _{x \rightarrow \infty} f(x)

\newline

Choose

1

answer:

\newline

(A)

2

\newline

(B)

3

\newline

(C)

0

\newline

(D) The limit is unbounded

Identify highest power of x: Identify the highest power of x in the numerator and the denominator. $\newline$ In the function $f(x) = \frac{6x^3}{3x + 2}$ , the highest power of x in the numerator is $x^3$ , and the highest power of x in the denominator is $x$ .
Divide terms by $x^3$ : Divide every term in the numerator and the denominator by $x^3$ , the highest power of $x$ in the numerator. $\newline$ $f(x) = \frac{6x^3/x^3}{(3x/x^3) + (2/x^3)}$ $\newline$ This simplifies to: $\newline$ $f(x) = \frac{6}{3/x^2 + 2/x^3}$
Evaluate limit at infinity: Evaluate the limit as $x$ approaches infinity. $\newline$ As $x$ approaches infinity, the terms $\frac{3}{x^2}$ and $\frac{2}{x^3}$ approach $0$ . $\newline$ So, the limit of $f(x)$ as $x$ approaches infinity is: $\newline$ $\lim_{x \to \infty} f(x) = \frac{6}{(0 + 0)} = \frac{6}{0}$
Recognize division by zero: Recognize that division by zero is undefined, which means the limit is unbounded. Since we cannot divide by $0$ , the limit of $f(x)$ as $x$ approaches $\infty$ does not exist or is unbounded.