Which statement best describes the limit shown below? lim_(x rarr oo)(-7x^(70))/(3x^(9)+10e^(x)) The limit equals zero The limit does not exist The limit exists and does not equal zero

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Which statement best describes the limit shown below?  lim_(x rarr oo)(-7x^(70))/(3x^(9)+10e^(x)) The limit equals zero The limit does not exist The limit exists and does not equal zero

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Given Limit Expression: We are given the limit expression lim_(x -> ∞)(-7x^(70))/(3x^(9)+10e^(x)). To find the behavior of this limit as x approaches infinity, we need to analyze the degrees of the polynomials in the numerator and the denominator, as well as the exponential function in the denominator.Degree Analysis: First, let's consider the highest power of x in both the numerator and the denominator. In the numerator, the highest power of x is x^(70), and in the denominator, the highest power of x is x^(9). The exponential function e^(x) grows faster than any polynomial, but we will first compare the polynomial terms.Dominant Term Comparison: Since the degree of x in the numerator (70) is much higher than the degree of x in the denominator (9), the polynomial part of the fraction will dominate the behavior of the limit as x approaches infinity. The exponential term 10e^(x) becomes insignificant compared to the x^(70) term as x grows larger.Simplification by Division: As x approaches infinity, the fraction (-7x^(70))/(3x^(9)) will dominate the behavior of the limit. We can simplify this fraction by dividing both the numerator and the denominator by x^(9), the highest power of x in the denominator.Exponential Term Behavior: After dividing by x^(9), we get (-7x^(61))/(3+10e^(x)/x^(9)). As x approaches infinity, the term 10e^(x)/x^(9) goes to zero because the exponential function e^(x) grows slower than x^(9). Therefore, the denominator approaches 3.Final Simplified Expression: Now, we are left with the simplified expression (-7x^(61))/3. As x approaches infinity, x^(61) also approaches infinity, and thus the whole expression approaches negative infinity.Limit Evaluation: Since the expression approaches negative infinity, the limit does not equal zero, and it does exist. Therefore, the correct statement is ＂The limit exists and does not equal zero.＂

Which statement best describes the limit shown below? $\newline$ $\lim _{x \rightarrow \infty} \frac{-7 x^{70}}{3 x^{9}+10 e^{x}}$ $\newline$ The limit equals zero $\newline$ The limit does not exist $\newline$ The limit exists and does not equal zero

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Which statement best describes the limit shown below?\newlinelim⁡x→∞−7x703x9+10ex \lim _{x \rightarrow \infty} \frac{-7 x^{70}}{3 x^{9}+10 e^{x}} x→∞lim​3x9+10ex−7x70​\newlineThe limit equals zero\newlineThe limit does not exist\newlineThe limit exists and does not equal zero

Which statement best describes the limit shown below? $\newline$ $\lim _{x \rightarrow \infty} \frac{-7 x^{70}}{3 x^{9}+10 e^{x}}$ $\newline$ The limit equals zero $\newline$ The limit does not exist $\newline$ The limit exists and does not equal zero