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)(13x^(3)+47x^(7)+x^(10)))/(7x+3x^(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)(13x^(3)+47x^(7)+x^(10)))/(7x+3x^(3)) Answer:

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Identify highest power: To find the limit of the given expression as x approaches infinity, we should first identify the highest power of x in both the numerator and the denominator to simplify the expression.Divide by x^(3): In the numerator, the highest power of x is x^(10) and in the denominator, the highest power of x is x^(3). To simplify, we can divide both the numerator and the denominator by x^(3), the highest power in the denominator.Simplify expression: After dividing by x^(3), the expression becomes:  lim_(x rarr oo)(root(3)(13x^(3)/x^(3) + 47x^(7)/x^(3) + x^(10)/x^(3)))/(7x/x^(3) + 3x^(3)/x^(3))  This simplifies to:  lim_(x rarr oo)(root(3)(13 + 47x^(4) + x^(7)))/(7/x^(2) + 3)Negligible terms: As x approaches infinity, the terms 13 and 3 become negligible compared to the terms with x. Also, 7/x^(2) approaches 0. So the expression further simplifies to:  lim_(x rarr oo)(root(3)(47x^(4) + x^(7)))/3Dominant term: Now, we can see that x^(7) is the dominant term in the cube root, so we can ignore the 47x^(4) term as x approaches infinity. The expression now simplifies to:  lim_(x rarr oo)(root(3)(x^(7)))/3Final limit: Taking the cube root of x^(7) gives us x^(7/3). So the expression is now:  lim_(x rarr oo)(x^(7/3))/3As x approaches infinity, x^(7/3) also approaches infinity. Therefore, the limit of the expression as x approaches infinity is infinity.

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{13 x^{3}+47 x^{7}+x^{10}}}{7 x+3 x^{3}}$ $\newline$ 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{13 x^{3}+47 x^{7}+x^{10}}}{7 x+3 x^{3}}$ $\newline$ Answer: