What is the limit as n approaches infinity of (n^{100})/(50^n) ?

The limit as n approaches infinity of (n^{100})/(50^n) is 0

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

limit as n approaches infinity of (n^{100})/(50^n)

n \to \infty lim (\frac{n ^{100}}{5 0 ^{n}})

0

Solution steps

n \to \infty lim (\frac{n ^{100}}{5 0 ^{n}})

Apply L'Hopital's Rule

= n \to \infty lim (\frac{100 !}{ln ^{100} ( 50 ) \cdot 5 0 ^{n}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{100 !}{ln ^{100} ( 50 )} \cdot n \to \infty lim (\frac{1}{5 0 ^{n}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{100 !}{ln ^{100} ( 50 )} \cdot \frac{lim _{n \to \infty} ( 1 )}{lim _{n \to \infty} ( 5 0 ^{n} )}

n \to \infty lim (1) = 1

n \to \infty lim (5 0^{n}) = \infty

= \frac{100 !}{ln ^{100} ( 50 )} \cdot \frac{1}{\infty}

Simplify

\frac{100 !}{ln ^{100} ( 50 )} \cdot \frac{1}{\infty} : 0

= 0

What is the limit as n approaches infinity of (n^{100})/(50^n) ?
The limit as n approaches infinity of (n^{100})/(50^n) is 0