Which statement best describes the limit shown below? lim_(x rarr oo)(-7ln x)/(3x^(38)+5x^(38)) The limit equals zero The limit does not exist The limit exists and does not equal zero

Identify Denominator: We are given the limit expression lim_(x -> ∞)(-7ln x)/(3x^(38)+5x^(38)). To find the limit as x approaches infinity, we need to analyze the behavior of the numerator and the denominator separately.Combine Like Terms: First, let's look at the denominator: 3x^(38) + 5x^(38). We can combine like terms by adding the coefficients of the x^(38) terms. 3x^(38) + 5x^(38) = (3 + 5)x^(38) = 8x^(38).Analyze Behavior: Now, we have the limit expression simplified to lim_(x -> ∞)(-7ln x)/(8x^(38)). The next step is to recognize that as x approaches infinity, the natural logarithm function ln x grows without bound, but at a much slower rate than any positive power of x.Recognize Growth Rates: Since the denominator contains x raised to the 38th power, it will grow much faster than the natural logarithm in the numerator as x approaches infinity. This means that the fraction as a whole will approach zero.Determine Limit: Therefore, the limit of (-7ln x)/(8x^(38)) as x approaches infinity is 0 because the denominator increases much faster than the numerator.

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Which statement best describes the limit shown below?

lim_(x rarr oo)(-7ln x)/(3x^(38)+5x^(38))
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? $\newline$ $\lim _{x \rightarrow \infty} \frac{-7 \ln x}{3 x^{38}+5 x^{38}}$ $\newline$ 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{-7 \ln x}{3 x^{38}+5 x^{38}}

\newline

The limit equals zero

\newline

The limit does not exist

\newline

The limit exists and does not equal zero

Identify Denominator: We are given the limit expression $\lim_{x \to \infty}\left(\frac{-7\ln x}{3x^{38}+5x^{38}}\right)$ . To find the limit as $x$ approaches infinity, we need to analyze the behavior of the numerator and the denominator separately.
Combine Like Terms: First, let's look at the denominator: $3x^{38} + 5x^{38}$ . We can combine like terms by adding the coefficients of the $x^{38}$ terms. $\newline$ $3x^{38} + 5x^{38} = (3 + 5)x^{38} = 8x^{38}.$
Analyze Behavior: Now, we have the limit expression simplified to $\lim_{x \to \infty}\left(\frac{-7\ln x}{8x^{38}}\right)$ . The next step is to recognize that as $x$ approaches infinity, the natural logarithm function $\ln x$ grows without bound, but at a much slower rate than any positive power of $x$ .
Recognize Growth Rates: Since the denominator contains $x$ raised to the $38^{\text{th}}$ power, it will grow much faster than the natural logarithm in the numerator as $x$ approaches infinity. This means that the fraction as a whole will approach zero.
Determine Limit: Therefore, the limit of $\frac{-7\ln x}{8x^{38}}$ as $x$ approaches infinity is $0$ because the denominator increases much faster than the numerator.