Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). lim_(x rarr oo)(sqrt(-19x^(3)+36x^(4)))/(3+x+x^(2)) Answer:

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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).  lim_(x rarr oo)(sqrt(-19x^(3)+36x^(4)))/(3+x+x^(2)) Answer:

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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. sqrt(-19x^(3)+36x^(4)) = x^2 * sqrt(-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. 3 + x + x^2 ≈ x^2Rewrite Original Function: Now we rewrite the original function, focusing on the dominant terms: (sqrt(-19x^(3)+36x^(4)))/(3+x+x^(2)) ≈ (x^2 * sqrt(-19/x + 36))/(x^2)Simplify x^2 Terms: We can now simplify the x^2 terms in the numerator and the denominator: (x^2 * sqrt(-19/x + 36))/(x^2) ≈ sqrt(-19/x + 36)Further Simplification: As x approaches infinity, the term -19/x approaches 0, so we can simplify the expression inside the square root further: sqrt(-19/x + 36) ≈ sqrt(36)Final Limit Calculation: The square root of 36 is 6, so the limit of the function as x approaches infinity is 6.

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:

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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).\newlinelim⁡x→∞−19x3+36x43+x+x2 \lim _{x \rightarrow \infty} \frac{\sqrt{-19 x^{3}+36 x^{4}}}{3+x+x^{2}} x→∞lim​3+x+x2−19x3+36x4​​\newlineAnswer:

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: