Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). lim_(x rarr oo)(root(3)(31-27x^(6)))/(7+8x+3x^(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)(root(3)(31-27x^(6)))/(7+8x+3x^(2)) Answer:

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Analyze Behavior: We are asked to find the limit of the function (root(3)(31-27x^(6)))/(7+8x+3x^(2)) as x approaches infinity. To do this, we will analyze the behavior of the numerator and the denominator separately as x grows without bound.Consider Highest Powers: First, let's consider the highest power of x in the numerator and the denominator. In the numerator, the highest power inside the cube root is x^6, and in the denominator, the highest power is x^2. To simplify the limit, we will divide both the numerator and the denominator by x^2, the highest power in the denominator.Divide by x^2: Dividing the numerator by x^2 inside the cube root gives us (root(3)(31/x^2 - 27x^4)). Dividing the denominator by x^2 gives us (7/x^2 + 8/x + 3).Simplify Numerator: As x approaches infinity, the terms 31/x^2 in the numerator and 7/x^2 and 8/x in the denominator approach 0. So, we can simplify the limit to the leading terms, which gives us the limit of (root(3)(-27x^4))/(3) as x approaches infinity.Take Cube Root: Now, we can take the cube root of -27x^4, which simplifies to -3x^(4/3). The limit now becomes the limit of (-3x^(4/3))/3 as x approaches infinity.Divide by 3: We can simplify the expression further by dividing -3x^(4/3) by 3, which gives us -x^(4/3). So, the limit we are trying to find is now the limit of -x^(4/3) as x approaches infinity.Simplify Further: Since x^(4/3) grows without bound as x approaches infinity, the limit of -x^(4/3) is negative infinity. Therefore, the limit does not exist.

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{31-27 x^{6}}}{7+8 x+3 x^{2}}$ $\newline$ 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{31-27 x^{6}}}{7+8 x+3 x^{2}}$ $\newline$ Answer: