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)(-50x^(6)+4+27x^(12)))/(x^(4)+4x^(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)(-50x^(6)+4+27x^(12)))/(x^(4)+4x^(2)) Answer:

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Observe Highest Powers: To find the limit of the given expression as x approaches infinity, we first observe the highest powers of x in the numerator and the denominator. In the numerator, the highest power is x^12 inside the cube root, which simplifies to x^4 outside the cube root. In the denominator, the highest power is x^4. We will divide both the numerator and the denominator by x^4 to simplify the expression.Divide by x^4: Divide each term in the numerator and the denominator by x^4: lim_(x rarr oo)(root(3)(-50x^(6)/x^(4) + 4/x^(4) + 27x^(12)/x^(4)))/(x^(4)/x^(4) + 4x^(2)/x^(4)) Simplify the expression: lim_(x rarr oo)(root(3)(-50x^(2) + 4/x^(4) + 27x^(8)))/(1 + 4/x^(2))Simplify Expression: As x approaches infinity, the terms with negative powers of x will approach zero. Therefore, 4/x^(4) in the numerator and 4/x^(2) in the denominator will become negligible: lim_(x rarr oo)(root(3)(-50x^(2) + 0 + 27x^(8)))/(1 + 0) Simplify the expression further: lim_(x rarr oo)(root(3)(27x^(8) - 50x^(2)))/1Approaching Infinity: Now, we can take the cube root of the terms inside the root separately since the limit of a sum is the sum of the limits: lim_(x rarr oo)(root(3)(27x^(8)) - root(3)(50x^(2))) Simplify the cube roots: lim_(x rarr oo)(3x^(8/3) - (50)^(1/3)x^(2/3))Take Cube Roots: As x approaches infinity, the term with the higher power of x will dominate. In this case, 3x^(8/3) will grow much faster than (50)^(1/3)x^(2/3), so the latter term becomes negligible: lim_(x rarr oo)(3x^(8/3))Dominant Term: The limit of 3x^(8/3) as x approaches infinity is infinity since the power of x is positive: lim_(x rarr oo)(3x^(8/3)) = oo Therefore, the limit does not exist (DNE).

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-50 x^{6}+4+27 x^{12}}}{x^{4}+4 x^{2}}$ $\newline$ 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→∞−50x6+4+27x123x4+4x2 \lim _{x \rightarrow \infty} \frac{\sqrt[3]{-50 x^{6}+4+27 x^{12}}}{x^{4}+4 x^{2}} x→∞lim​x4+4x23−50x6+4+27x12​​\newlineAnswer:

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-50 x^{6}+4+27 x^{12}}}{x^{4}+4 x^{2}}$ $\newline$ Answer: