What is the limit as n approaches infinity of n/(6^n) ?

The limit as n approaches infinity of n/(6^n) is 0

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limit as n approaches infinity of n/(6^n)

n \to \infty lim (\frac{n}{6 ^{n}})

0

Solution steps

n \to \infty lim (\frac{n}{6 ^{n}})

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{n}{6 ^{n}} = (n) 6^{- n}

= n \to \infty lim (n \cdot 6^{- n})

Rewrite for L'Hopital

= n \to \infty lim (\frac{6 ^{- n}}{\frac{1}{n}})

Apply L'Hopital's Rule

: n \to \infty lim (\frac{- ln ( 6 ) \cdot 6 ^{- n}}{- \frac{1}{n ^{2}}})

= n \to \infty lim (\frac{- ln ( 6 ) \cdot 6 ^{- n}}{- \frac{1}{n ^{2}}})

Simplify

= n \to \infty lim (ln (6) \cdot 6^{- n} n^{2})

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

= ln (6) \cdot n \to \infty lim (6^{- n} n^{2})

Use the series convergence definition

: 0

= ln (6) \cdot 0

Simplify

= 0

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f (x) = x + \frac{4}{x}, (4, 5)

What is the limit as n approaches infinity of n/(6^n) ?
The limit as n approaches infinity of n/(6^n) is 0