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)(48x^(5)+x^(6)-39x^(2)))/(10x^(2)+7x^(3)) 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)(48x^(5)+x^(6)-39x^(2)))/(10x^(2)+7x^(3)) Answer:

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Simplify expression by factoring: We are given the limit: lim_(x -> ∞)(root(3)(48x^(5)+x^(6)-39x^(2)))/(10x^(2)+7x^(3))  Step 1: Simplify the expression inside the cube root by factoring out the highest power of x common to all terms.Focus on highest power terms: Since x is approaching infinity, we can focus on the terms with the highest power of x in the numerator and the denominator.  In the numerator, the term with the highest power of x is x^6, and in the denominator, it is x^3.  We can factor x^5 out of the cube root in the numerator to simplify the expression. root(3)(48x^(5)+x^(6)-39x^(2)) = root(3)(x^5(x + 48 - 39/x^3))Factor out highest power of x: Simplify the expression in the denominator by factoring out the highest power of x common to all terms. 10x^(2)+7x^(3) = x^2(10 + 7x)Rewrite limit with simplified expressions: Now we rewrite the limit with the simplified expressions. lim_(x -> ∞)(root(3)(x^5(x + 48 - 39/x^3)))/(x^2(10 + 7x))Divide by x^2: We can now divide both the numerator and the denominator by x^2. lim_(x -> ∞)(root(3)(x^5(x + 48 - 39/x^3))/x^2)/(10 + 7x)Simplify expression inside cube root: Simplify the expression inside the cube root by canceling out x^2. root(3)(x^5(x + 48 - 39/x^3))/x^2 = root(3)(x^3(x + 48 - 39/x^3))Take x out of cube root: Since we have a cube root, we can take x^3 out of the cube root. root(3)(x^3(x + 48 - 39/x^3)) = x * root(3)(x + 48 - 39/x^3)Rewrite limit with x taken out: Now we rewrite the limit with the x taken out of the cube root. lim_(x -> ∞)(x * root(3)(x + 48 - 39/x^3))/(10 + 7x)Negligible terms as x approaches infinity: As x approaches infinity, the terms 48 and -39/x^3 inside the cube root become negligible compared to x. Similarly, the term 10 in the denominator becomes negligible compared to 7x. lim_(x -> ∞)(x * root(3)(x))/(7x)Cancel out x in numerator and denominator: Simplify the expression by canceling out x in the numerator and denominator. lim_(x -> ∞)(root(3)(x))/7Cube root of x approaches infinity: As x approaches infinity, the cube root of x also approaches infinity. lim_(x -> ∞)(root(3)(x))/7 = ∞/7Limit of constant divided by infinity: The limit of a constant divided by infinity is 0. lim_(x -> ∞)(root(3)(x))/7 = 0

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{48 x^{5}+x^{6}-39 x^{2}}}{10 x^{2}+7 x^{3}}$ $\newline$ Answer:

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