What is the limit as x approaches infinity of xsin((4pi)/x) ?

The limit as x approaches infinity of xsin((4pi)/x) is 4pi

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limit as x approaches infinity of xsin((4pi)/x)

x \to \infty lim (x sin (\frac{4 π}{x}))

4 π

Decimal Notation

12.56637 \dots

Solution steps

x \to \infty lim (x sin (\frac{4 π}{x}))

Rewrite for L'Hopital

= x \to \infty lim (\frac{sin ( \frac{4 π}{x} )}{\frac{1}{x}})

Apply L'Hopital's Rule

: x \to \infty lim \frac{- \frac{4 π c o s ( \frac{4 π}{x} )}{x ^{2}}}{- \frac{1}{x ^{2}}}

= x \to \infty lim \frac{- \frac{4 π c o s ( \frac{4 π}{x} )}{x ^{2}}}{- \frac{1}{x ^{2}}}

Simplify

\frac{- \frac{4 π c o s ( \frac{4 π}{x} )}{x ^{2}}}{- \frac{1}{x ^{2}}} : 4 π cos (\frac{4 π}{x})

= x \to \infty lim (4 π cos (\frac{4 π}{x}))

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

= 4 π \cdot x \to \infty lim (cos (\frac{4 π}{x}))

Apply the Limit Chain Rule:

1

= 4 π 1

Simplify

= 4 π

\frac{\partial}{\partial z} (x^{2})

sin^{2} (t)

f (x) = 2 x^{3} - x^{2} + 9, (2, 21)

\int \frac{4}{35 sin ( x ) - 12 cos ( x )} d x

y^{^{''}} - 2 y^{^{'}} - 3 y = - 9 t e^{- t}

What is the limit as x approaches infinity of xsin((4pi)/x) ?
The limit as x approaches infinity of xsin((4pi)/x) is 4pi