Which statement best describes the limit shown below? lim_(x rarr oo)(x^(68))/(6x^(68)+x^(62)) The limit equals zero The limit does not exist The limit exists and does not equal zero

Analyze Degree Comparison: We are given the limit expression lim_(x -> ∞)(x^(68))/(6x^(68) + x^(62)). To find the limit as x approaches infinity, we can compare the degrees of the polynomials in the numerator and the denominator.Divide by Highest Degree Term: Since the highest degree term in both the numerator and the denominator is x^(68), we can divide every term by x^(68) to simplify the expression. lim_(x -> ∞)(x^(68))/(6x^(68) + x^(62)) = lim_(x -> ∞)(1)/(6 + x^(62)/x^(68))Simplify Expression: Simplify the expression inside the limit by canceling out x^(62) from the denominator's second term. lim_(x -> ∞)(1)/(6 + x^(62)/x^(68)) = lim_(x -> ∞)(1)/(6 + 1/x^(6))Negligible Term Removal: As x approaches infinity, 1/x^(6) approaches 0. Therefore, the term 1/x^(6) in the denominator becomes negligible. lim_(x -> ∞)(1)/(6 + 1/x^(6)) = lim_(x -> ∞)(1)/(6 + 0)Final Simplification: Simplify the expression by removing the term that approaches zero. lim_(x -> ∞)(1)/(6 + 0) = 1/6Limit Result: The limit of the function as x approaches infinity is 1/6, which is a finite number and does not equal zero.

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Which statement best describes the limit shown below?

lim_(x rarr oo)(x^(68))/(6x^(68)+x^(62))
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? $\newline$ $\lim _{x \rightarrow \infty} \frac{x^{68}}{6 x^{68}+x^{62}}$ $\newline$ 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^{68}}{6 x^{68}+x^{62}}

\newline

The limit equals zero

\newline

The limit does not exist

\newline

The limit exists and does not equal zero

Analyze Degree Comparison: We are given the limit expression $\lim_{x \to \infty}\left(\frac{x^{68}}{6x^{68} + x^{62}}\right)$ . To find the limit as $x$ approaches infinity, we can compare the degrees of the polynomials in the numerator and the denominator.
Divide by Highest Degree Term: Since the highest degree term in both the numerator and the denominator is $x^{68}$ , we can divide every term by $x^{68}$ to simplify the expression. $\newline$ $\lim_{x \to \infty}\frac{x^{68}}{6x^{68} + x^{62}} = \lim_{x \to \infty}\frac{1}{6 + \frac{x^{62}}{x^{68}}}$
Simplify Expression: Simplify the expression inside the limit by canceling out $x^{62}$ from the denominator's second term. $\newline$ $\lim_{x \to \infty}\left(\frac{1}{6 + \frac{x^{62}}{x^{68}}}\right) = \lim_{x \to \infty}\left(\frac{1}{6 + \frac{1}{x^{6}}}\right)$
Negligible Term Removal: As $x$ approaches infinity, $\frac{1}{x^{6}}$ approaches $0$ . Therefore, the term $\frac{1}{x^{6}}$ in the denominator becomes negligible. $\newline$ $\lim_{x \to \infty}\frac{1}{6 + \frac{1}{x^{6}}} = \lim_{x \to \infty}\frac{1}{6 + 0}$
Final Simplification: Simplify the expression by removing the term that approaches zero. $\newline$ $\lim_{x \to \infty}(\frac{1}{6 + 0}) = \frac{1}{6}$
Limit Result: The limit of the function as $x$ approaches infinity is $\frac{1}{6}$ , which is a finite number and does not equal zero.