For which of the following does \(\lim\limits_{x\to \infty} f(x)=0 \) ?

• \(\hskip{1em} f(x)=\dfrac{\ln x}{x^{100}}\)
• \(\hskip{1em} f(x)=\dfrac{e^x}{\ln x}\)
• \(\hskip{1em} f(x)=\dfrac{x^{100}}{e^x}\)

Summary Submit

We are looking for functions where the denominators grow faster than the numerators.

Natural log functions grow slower than power and exponential functions. Exponential functions grow faster than power functions.

$$ e^x \gg x^n \gg \ln x \text{ for large } x \text{ values}$$

For this question, the options where the denominator grows faster than the numerator will result in the limit approaching \(0\).