e$.$ The functions $ln{x}^{100}$ and $lnx$ have comparable growth rates as $x.$ Is the statement true or false? Explain.

A$.$ True; by l$\text{'}$H$$pital$\text{'}$s Rule, $\underset{x}{lim}\frac{ln{x}^{100}}{lnx}=\underset{x}{lim}\frac{\frac{1}{x^{100}}*100x^{99}}{\frac{1}{x}}=\underset{x}{lim}100=100.$

B$.$ False; by l$\text{'}$H$$pital$\text{'}$s Rule, $\underset{x}{lim}\frac{ln{x}^{100}}{lnx}=\underset{x}{lim}\frac{\frac{1}{x^{100}}*100x^{99}}{\frac{1}{x}}=\underset{x}{lim}x=.$

C$.$ False; since $\frac{ln{x}^{100}}{lnx}$ is in the indeterminate form $\frac{}{},$ the functions do not have comparable growth rates.

D$.$ False: $\underset{x}{lim}\frac{ln{x}^{100}}{lnx}=ln{x}^{99}=.$