Question - Behavior of Exponential and Polynomial Functions in Limits
Solution:
The limit presented in the image is:\[ \lim_{{x \to \infty}} \frac{3e^x - 2}{x^3} \]To solve this, we need to assess the behavior of the numerator and the denominator as x approaches infinity.The term $$3e^x$$ grows exponentially as x approaches infinity, while the $$x^3$$ term grows at a cubic rate. Since exponential growth (in this case $$e^x$$) outpaces any polynomial growth ($$x^3$$), the numerator will become infinitely larger compared to the denominator.Therefore, as $$x$$ approaches infinity, the fraction $$\frac{3e^x - 2}{x^3}$$ will approach infinity. Thus, the limit is:\[ \lim_{{x \to \infty}} \frac{3e^x - 2}{x^3} = \infty \]
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