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

The limit as n approaches infinity of n/6 sin(6/n) is 1

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

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

1

Solution steps

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

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

= \frac{1}{6} \cdot n \to \infty lim (n sin (\frac{6}{n}))

Rewrite for L'Hopital

= \frac{1}{6} \cdot n \to \infty lim (\frac{sin ( \frac{6}{n} )}{\frac{1}{n}})

Apply L'Hopital's Rule

: n \to \infty lim \frac{- \frac{6 c o s ( \frac{6}{n} )}{n ^{2}}}{- \frac{1}{n ^{2}}}

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

Simplify

\frac{- \frac{6 c o s ( \frac{6}{n} )}{n ^{2}}}{- \frac{1}{n ^{2}}} : 6 cos (\frac{6}{n})

= \frac{1}{6} \cdot n \to \infty lim (6 cos (\frac{6}{n}))

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

= \frac{1}{6} \cdot 6 \cdot n \to \infty lim (cos (\frac{6}{n}))

Apply the Limit Chain Rule:

1

= \frac{1}{6} \cdot 6 \cdot 1

Simplify

\frac{1}{6} \cdot 6 \cdot 1 : 1

= 1

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What is the limit as n approaches infinity of n/6 sin(6/n) ?
The limit as n approaches infinity of n/6 sin(6/n) is 1