Which statement best describes the limit shown below? lim_(x rarr oo)(x^(37))/(log_(4)x+4^(x)) The limit equals zero The limit does not exist The limit exists and does not equal zero

Analyze Function Behavior: We need to analyze the behavior of the function (x^(37))/(log_4(x) + 4^x) as x approaches infinity. To do this, we will compare the growth rates of the numerator and the denominator.Numerator Growth Rate: The numerator x^(37) is a polynomial function of degree 37, which grows very quickly as x becomes large.Denominator Comparison: The denominator is the sum of log_4(x) and 4^x. The logarithmic function log_4(x) grows slowly compared to the exponential function 4^x. Therefore, as x approaches infinity, the 4^x term will dominate the denominator.Dominant Term: Since the exponential function 4^x grows much faster than the polynomial function x^(37), the denominator will increase much more rapidly than the numerator as x approaches infinity.Approaching Infinity: As a result, the fraction (x^(37))/(log_4(x) + 4^x) will approach zero because the numerator becomes insignificant compared to the rapidly increasing denominator.Limit Statement: Therefore, the correct statement that describes the limit is: ＂The limit equals zero.＂

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Which statement best describes the limit shown below?

lim_(x rarr oo)(x^(37))/(log_(4)x+4^(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? $\newline$ $\lim _{x \rightarrow \infty} \frac{x^{37}}{\log _{4} x+4^{x}}$ $\newline$ 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{x^{37}}{\log _{4} x+4^{x}}

\newline

The limit equals zero

\newline

The limit does not exist

\newline

The limit exists and does not equal zero

Analyze Function Behavior: We need to analyze the behavior of the function $\frac{x^{37}}{\log_{4}(x) + 4^{x}}$ as $x$ approaches infinity. To do this, we will compare the growth rates of the numerator and the denominator.
Numerator Growth Rate: The numerator $x^{37}$ is a polynomial function of degree $37$ , which grows very quickly as $x$ becomes large.
Denominator Comparison: The denominator is the sum of $\log_4(x)$ and $4^x$ . The logarithmic function $\log_4(x)$ grows slowly compared to the exponential function $4^x$ . Therefore, as $x$ approaches infinity, the $4^x$ term will dominate the denominator.
Dominant Term: Since the exponential function $4^x$ grows much faster than the polynomial function $x^{37}$ , the denominator will increase much more rapidly than the numerator as $x$ approaches infinity.
Approaching Infinity: As a result, the fraction $(x^{37})/(\log_4(x) + 4^x)$ will approach zero because the numerator becomes insignificant compared to the rapidly increasing denominator.
Limit Statement: Therefore, the correct statement that describes the limit is: ＂The limit equals $0$ .＂