Which statement best describes the limit shown below? lim_(x rarr oo)(-6log_(5)x+x^(89))/(x^(18)+5^(x)) The limit equals zero The limit does not exist The limit exists and does not equal zero

Analyze Behavior as x Approaches Infinity: We need to analyze the behavior of the numerator and the denominator as x approaches infinity. The numerator is -6log_(5)x + x^(89), and the denominator is x^(18) + 5^(x). As x approaches infinity, x^(89) will grow much faster than -6log_(5)x, so the dominant term in the numerator is x^(89). Similarly, in the denominator, 5^(x) will grow much faster than x^(18), so the dominant term in the denominator is 5^(x).Compare Growth Rates of Dominant Terms: To determine the limit, we compare the growth rates of the dominant terms in the numerator and the denominator. The term x^(89) is a polynomial term, and it grows faster than any logarithmic function but slower than an exponential function like 5^(x). Therefore, as x approaches infinity, the denominator's growth rate will outpace the numerator's growth rate.Determine Limit Approach: Since the denominator grows faster than the numerator, the fraction as a whole will approach zero as x approaches infinity. This means that the limit of the function as x approaches infinity is zero.

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Which statement best describes the limit shown below?

lim_(x rarr oo)(-6log_(5)x+x^(89))/(x^(18)+5^(x))
The limit equals zero
The limit does not exist
The limit exists and does not equal zero

Which statement best describes the limit shown below? $\newline$ $\lim _{x \rightarrow \infty} \frac{-6 \log _{5} x+x^{89}}{x^{18}+5^{x}}$ $\newline$ The limit equals zero $\newline$ The limit does not exist $\newline$ The limit exists and does not equal zero

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Q. Which statement best describes the limit shown below?

\newline

\lim _{x \rightarrow \infty} \frac{-6 \log _{5} x+x^{89}}{x^{18}+5^{x}}

\newline

The limit equals zero

\newline

The limit does not exist

\newline

The limit exists and does not equal zero

Analyze Behavior as $x$ Approaches Infinity: We need to analyze the behavior of the numerator and the denominator as $x$ approaches infinity. $\newline$ The numerator is $-6\log_{5}x + x^{89}$ , and the denominator is $x^{18} + 5^{x}$ . $\newline$ As $x$ approaches infinity, $x^{89}$ will grow much faster than $-6\log_{5}x$ , so the dominant term in the numerator is $x^{89}$ . $\newline$ Similarly, in the denominator, $5^{x}$ will grow much faster than $x^{18}$ , so the dominant term in the denominator is $5^{x}$ .
Compare Growth Rates of Dominant Terms: To determine the limit, we compare the growth rates of the dominant terms in the numerator and the denominator. $\newline$ The term $x^{89}$ is a polynomial term, and it grows faster than any logarithmic function but slower than an exponential function like $5^{x}$ . $\newline$ Therefore, as $x$ approaches infinity, the denominator's growth rate will outpace the numerator's growth rate.
Determine Limit Approach: Since the denominator grows faster than the numerator, the fraction as a whole will approach $0$ as $x$ approaches infinity. This means that the limit of the function as $x$ approaches infinity is $0$ .