What is the limit as x approaches infinity of [xsin(1/x)] ?

The limit as x approaches infinity of [xsin(1/x)] is 1

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limit as x approaches infinity of [xsin(1/x)]

x \to \infty lim ([x sin (\frac{1}{x})])

1

Solution steps

x \to \infty lim (x sin (\frac{1}{x}))

Rewrite for L'Hopital

= x \to \infty lim (\frac{sin ( \frac{1}{x} )}{\frac{1}{x}})

Apply L'Hopital's Rule

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

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

Simplify

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

= x \to \infty lim (cos (\frac{1}{x}))

Apply the Limit Chain Rule:

1

= 1

x \to (0) lim (\frac{1 - 1 + x y}{x y})

x \to 0 lim (+ \frac{1}{x})

x \to 3 lim (\frac{3 - 12 - x}{x - 3})

x \to 0 lim (\frac{1}{cos ( 6 x )})

x \to 1 lim (4 arcsin (x))

What is the limit as x approaches infinity of [xsin(1/x)] ?
The limit as x approaches infinity of [xsin(1/x)] is 1