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^(12)-23x^(8)))/(6x^(3)+9) Answer:

Question

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^(12)-23x^(8)))/(6x^(3)+9) Answer:

Bytelearn · Accepted Answer

Identify Powers of x: Identify the highest power of x in both the numerator and the denominator. In the numerator, the highest power of x inside the cube root is x^12, and in the denominator, the highest power of x is x^3.Factor Out Highest Power: Factor out the highest power of x from the cube root in the numerator. Since the highest power of x inside the cube root is x^12, we can factor out x^4 from the cube root (because (x^4)^3 = x^12). The expression becomes: lim_(x rarr oo)(root(3)(x^12(27-23/x^4)))/(6x^3+9)Simplify Cube Root: Simplify the cube root in the numerator. The cube root of x^12 is x^4, so the expression simplifies to: lim_(x rarr oo)(x^4 * root(3)(27-23/x^4))/(6x^3+9)Divide by x^3: Divide every term by x^3, the highest power of x in the denominator. This gives us: lim_(x rarr oo)((x^4/x^3) * root(3)(27-23/x^4))/(6+9/x^3)Cancel x^3 in Numerator: Simplify the expression by canceling out x^3 from x^4 in the numerator. This simplifies to: lim_(x rarr oo)(x * root(3)(27-23/x^4))/(6+9/x^3)Evaluate Limit at Infinity: Evaluate the limit as x approaches infinity. As x approaches infinity, 23/x^4 approaches 0 and 9/x^3 approaches 0. The expression simplifies to: lim_(x rarr oo)(x * root(3)(27))/(6)Calculate Cube Root: Calculate the cube root of 27 and simplify the expression. The cube root of 27 is 3, so the expression simplifies to: lim_(x rarr oo)(3x)/6Divide by 6: Simplify the expression by dividing 3x by 6. This simplifies to: lim_(x rarr oo)(x/2)Evaluate Limit at Infinity: Evaluate the limit as x approaches infinity. As x approaches infinity, x/2 also approaches infinity. 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{12}-23 x^{8}}}{6 x^{3}+9}$ $\newline$ Answer:

Full solution

More problems from Find derivatives of logarithmic functions

Related topics

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).\newlinelim⁡x→∞27x12−23x836x3+9 \lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{12}-23 x^{8}}}{6 x^{3}+9} x→∞lim​6x3+9327x12−23x8​​\newlineAnswer:

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{12}-23 x^{8}}}{6 x^{3}+9}$ $\newline$ Answer: