Prove that the exponential function grows faster than any power function by evaluating the limit of their ratio at infinity.

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Limit of x^n / e^x as x approaches infinity

Logarithmic and exponential functions core

Prove that, for any positive \(n\), \(\lim_{x \to +\infty} \frac{x^n}{e^x} = 0\).

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What you need first

Properties of exponents and logarithms
The limit of ln(x)/x as x approaches infinity
The limit of e^x as x approaches infinity

What this problem tests

Tests the ability to manipulate exponential and logarithmic expressions to evaluate limits without relying on repeated applications of L'Hôpital's rule.

This is a classic result in Calculus I, often used to establish the hierarchy of growth rates for functions at infinity.

Limit of x^n / e^x as x approaches infinity

What you need first

What this problem tests

More on Logarithmic and exponential functions