Which statement best describes the limit shown below? lim_(x rarr oo)(x^(19))/(2x^(19)-10e^(x)) The limit equals zero The limit does not exist The limit exists and does not equal zero

Analyze Highest Degree Terms: We are given the limit expression lim_(x -> ∞)(x^(19))/(2x^(19) - 10e^(x)). To find the behavior of this function as x approaches infinity, we need to analyze the highest degree terms in the numerator and the denominator.Exponential Term Dominance: In the numerator, we have x^(19), and in the denominator, we have 2x^(19) and -10e^(x). As x approaches infinity, the exponential term e^(x) grows much faster than any polynomial term, including x^(19).Denominator Control: Since e^(x) grows faster than x^(19), the term -10e^(x) in the denominator will dominate over 2x^(19) as x approaches infinity. This means that the denominator will essentially be controlled by the -10e^(x) term.Approaching Zero: Given that the exponential term dominates and grows without bound, the fraction x^(19)/(2x^(19) - 10e^(x)) will approach zero as x approaches infinity. This is because the numerator grows at a polynomial rate, while the denominator grows at an exponential rate.Limit Statement: Therefore, the correct statement that describes the limit is: ＂The limit equals zero.＂

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Which statement best describes the limit shown below?

lim_(x rarr oo)(x^(19))/(2x^(19)-10e^(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? $\newline$ $\lim _{x \rightarrow \infty} \frac{x^{19}}{2 x^{19}-10 e^{x}}$ $\newline$ The limit equals zero $\newline$ The limit does not exist $\newline$ The limit exists and does not equal zero

View step-by-step help

Full solution

Q. Which statement best describes the limit shown below?

\newline

\lim _{x \rightarrow \infty} \frac{x^{19}}{2 x^{19}-10 e^{x}}

\newline

The limit equals zero

\newline

The limit does not exist

\newline

The limit exists and does not equal zero

Analyze Highest Degree Terms: We are given the limit expression $\lim_{x \to \infty}\left(\frac{x^{19}}{2x^{19} - 10e^{x}}\right)$ . To find the behavior of this function as $x$ approaches infinity, we need to analyze the highest degree terms in the numerator and the denominator.
Exponential Term Dominance: In the numerator, we have $x^{19}$ , and in the denominator, we have $2x^{19}$ and $-10e^{x}$ . As $x$ approaches infinity, the exponential term $e^{x}$ grows much faster than any polynomial term, including $x^{19}$ .
Denominator Control: Since $e^{x}$ grows faster than $x^{19}$ , the term $-10e^{x}$ in the denominator will dominate over $2x^{19}$ as $x$ approaches infinity. This means that the denominator will essentially be controlled by the $-10e^{x}$ term.
Approaching Zero: Given that the exponential term dominates and grows without bound, the fraction $\frac{x^{19}}{2x^{19} - 10e^{x}}$ will approach zero as $x$ approaches infinity. This is because the numerator grows at a polynomial rate, while the denominator grows at an exponential rate.
Limit Statement: Therefore, the correct statement that describes the limit is: ＂The limit equals $0$ .＂