Which statement best describes the limit shown below? lim_(x rarr oo)(x^(71)+log_(3)x)/(3x^(71)+x^(29)) The limit equals zero The limit does not exist The limit exists and does not equal zero

Given Limit Analysis: We are given the limit: lim_(x -> ∞)(x^(71)+log_(3)x)/(3x^(71)+x^(29)) To find the limit as x approaches infinity, we need to analyze the behavior of the numerator and the denominator separately.Highest Power Comparison: First, let's consider the highest power of x in both the numerator and the denominator, which will dominate the behavior of the function as x approaches infinity. In the numerator, the highest power of x is x^(71). In the denominator, the highest power of x is 3x^(71).Simplification by Division: We can divide both the numerator and the denominator by x^(71) to simplify the expression. This gives us: lim_(x -> ∞)((x^(71)/x^(71))+(log_(3)x)/x^(71))/((3x^(71)/x^(71))+(x^(29)/x^(71)))Limit of Individual Terms: Simplifying the powers of x, we get: lim_(x -> ∞)(1+(log_(3)x)/x^(71))/(3+(x^(29)/x^(71)))Final Simplification: As x approaches infinity, (log_(3)x)/x^(71) approaches 0 because the logarithmic function grows much slower than the power function x^(71). Similarly, (x^(29)/x^(71)) also approaches 0 because the exponent in the numerator is much smaller than the exponent in the denominator.After taking the limits of the individual terms, we get: lim_(x -> ∞)(1+0)/(3+0)Simplifying the expression, we find that the limit is: lim_(x -> ∞)(1)/(3)The limit of a constant is the constant itself, so the limit is: 1/3

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Which statement best describes the limit shown below?

lim_(x rarr oo)(x^(71)+log_(3)x)/(3x^(71)+x^(29))
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? $\newline$ $\lim _{x \rightarrow \infty} \frac{x^{71}+\log _{3} x}{3 x^{71}+x^{29}}$ $\newline$ 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^{71}+\log _{3} x}{3 x^{71}+x^{29}}

\newline

The limit equals zero

\newline

The limit does not exist

\newline

The limit exists and does not equal zero

Given Limit Analysis: We are given the limit: $\newline$ $\lim_{x \to \infty}\frac{x^{71}+\log_{3}x}{3x^{71}+x^{29}}$ $\newline$ To find the limit as $x$ approaches infinity, we need to analyze the behavior of the numerator and the denominator separately.
Highest Power Comparison: First, let's consider the highest power of $x$ in both the numerator and the denominator, which will dominate the behavior of the function as $x$ approaches infinity. $\newline$ In the numerator, the highest power of $x$ is $x^{71}$ . $\newline$ In the denominator, the highest power of $x$ is $3x^{71}$ .
Simplification by Division: We can divide both the numerator and the denominator by $x^{71}$ to simplify the expression. $\newline$ This gives us: $\newline$ $\lim_{x \to \infty}\left(\frac{x^{71}}{x^{71}}+\frac{\log_{3}x}{x^{71}}\right)/\left(\frac{3x^{71}}{x^{71}}+\frac{x^{29}}{x^{71}}\right)$
Limit of Individual Terms: Simplifying the powers of x, we get: $\lim_{x \to \infty}\left(\frac{1+\left(\log_{3}x\right)/x^{71}}{3+\left(x^{29}/x^{71}\right)}\right)$
Final Simplification: As $x$ approaches infinity, $\frac{\log_{3}x}{x^{71}}$ approaches $0$ because the logarithmic function grows much slower than the power function $x^{71}$ . $\newline$ Similarly, $\frac{x^{29}}{x^{71}}$ also approaches $0$ because the exponent in the numerator is much smaller than the exponent in the denominator.
Final Simplification: As $x$ approaches infinity, $\frac{\log_{3}x}{x^{71}}$ approaches $0$ because the logarithmic function grows much slower than the power function $x^{71}$ . $\newline$ Similarly, $\frac{x^{29}}{x^{71}}$ also approaches $0$ because the exponent in the numerator is much smaller than the exponent in the denominator.After taking the limits of the individual terms, we get: $\newline$ $\lim_{x \to \infty}\frac{1+0}{3+0}$
Final Simplification: As $x$ approaches infinity, $\frac{\log_{3}x}{x^{71}}$ approaches $0$ because the logarithmic function grows much slower than the power function $x^{71}$ . $\newline$ Similarly, $\frac{x^{29}}{x^{71}}$ also approaches $0$ because the exponent in the numerator is much smaller than the exponent in the denominator.After taking the limits of the individual terms, we get: $\newline$ $\lim_{x \to \infty}\frac{1+0}{3+0}$ Simplifying the expression, we find that the limit is: $\newline$ $\lim_{x \to \infty}\frac{1}{3}$
Final Simplification: As $x$ approaches infinity, $\frac{\log_{3}x}{x^{71}}$ approaches $0$ because the logarithmic function grows much slower than the power function $x^{71}$ . Similarly, $\frac{x^{29}}{x^{71}}$ also approaches $0$ because the exponent in the numerator is much smaller than the exponent in the denominator.After taking the limits of the individual terms, we get: $\lim_{x \to \infty}\frac{1+0}{3+0}$ Simplifying the expression, we find that the limit is: $\lim_{x \to \infty}\frac{1}{3}$ The limit of a constant is the constant itself, so the limit is: $\frac{1}{3}$