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)(27x^(6)-14x^(3)))/(7+7x^(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)(27x^(6)-14x^(3)))/(7+7x^(2)) Answer:

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Factor out highest power: We are given the limit expression: lim_(x → ∞)(root(3)(27x^(6)-14x^(3)))/(7+7x^(2)) To simplify this expression, we will first factor out the highest power of x in the numerator and denominator.Factor out x^6: In the numerator, the highest power of x inside the cube root is x^6. We factor out x^6 from each term inside the cube root: root(3)(27x^(6)-14x^(3)) = root(3)(x^(6)(27-14/x^(3)))Take x^2 outside: Now we take x^(6) outside of the cube root, remembering that when we take a power out of a root, we divide the exponent by the root index: root(3)(x^(6)(27-14/x^(3))) = x^(6/3) * root(3)(27-14/x^(3)) = x^2 * root(3)(27-14/x^(3))Factor out x^2: In the denominator, we factor out x^2 from each term: 7+7x^(2) = 7(1+x^(2))Rewrite with factored terms: Now we rewrite the limit expression with the factored terms: lim_(x → ∞)(x^2 * root(3)(27-14/x^(3)))/(7(1+x^(2)))Divide by x^2: We can now divide both the numerator and the denominator by x^2: lim_(x → ∞)(x^2/x^2 * root(3)(27-14/x^(3)))/(7(1+x^(2))/x^2)Simplify expression: Simplifying the expression by canceling out x^2 in the numerator and dividing each term in the denominator by x^2 gives us: lim_(x → ∞)(root(3)(27-14/x^(3)))/(7/x^2+7)Approach infinity: As x approaches infinity, 14/x^(3) approaches 0 and 7/x^2 also approaches 0. Therefore, the limit simplifies to: lim_(x → ∞)(root(3)(27))/(7)Simplify further: The cube root of 27 is 3, so the limit simplifies further to: lim_(x → ∞)(3)/(7)Final limit: Since 3/7 is a constant, the limit as x approaches infinity is simply 3/7.

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{6}-14 x^{3}}}{7+7 x^{2}}$ $\newline$ 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{6}-14 x^{3}}}{7+7 x^{2}}$ $\newline$ Answer: