Behavior of Exponential and Polynomial Functions in Limits
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:
\[ \lim_{{x \to \infty}} \frac{3e^x - 2}{x^3} = \infty \]
