Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{-19 x^{3}+36 x^{4}}}{3+x+x^{2}}$ $\newline$ Answer:

View step-by-step help

Home

Math Problems

Precalculus

Find limits of polynomials and rational functions

Full solution

Q. Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).

\newline

\lim _{x \rightarrow \infty} \frac{\sqrt{-19 x^{3}+36 x^{4}}}{3+x+x^{2}}

\newline

Answer:

Analyze Behavior of Functions: To find the limit of the function as $x$ approaches infinity, we need to analyze the behavior of the numerator and the denominator separately. We will start by simplifying the function, focusing on the highest power of $x$ in both the numerator and the denominator.
Simplify Numerator: In the numerator, we have a square root of a polynomial. The highest power of $x$ inside the square root is $x^4$ , so we can factor out $x^2$ from the square root to simplify the expression. $\newline$ $\sqrt{-19x^{3}+36x^{4}} = x^2 \cdot \sqrt{-\frac{19}{x} + 36}$
Focus on Denominator: In the denominator, the highest power of $x$ is $x^2$ . As $x$ approaches infinity, the lower powers of $x$ ( $3$ and $x$ ) become insignificant compared to $x^2$ . Therefore, we can focus on $x^2$ in the denominator. $\newline$ $3 + x + x^2 \approx x^2$
Rewrite Original Function: Now we rewrite the original function, focusing on the dominant terms: $\frac{\sqrt{-19x^{3}+36x^{4}}}{3+x+x^{2}} \approx \frac{x^2 \sqrt{-\frac{19}{x} + 36}}{x^2}$
Simplify $x^2$ Terms: We can now simplify the $x^2$ terms in the numerator and the denominator: $\frac{x^2 \sqrt{-\frac{19}{x} + 36}}{x^2} \approx \sqrt{-\frac{19}{x} + 36}$
Further Simplification: As $x$ approaches infinity, the term $-\frac{19}{x}$ approaches $0$ , so we can simplify the expression inside the square root further: $\sqrt{-\frac{19}{x} + 36} \approx \sqrt{36}$
Final Limit Calculation: The square root of $36$ is $6$ , so the limit of the function as $x$ approaches infinity is $6$ .