What is the limit as x approaches infinity of x*sin(7/x) ?

The limit as x approaches infinity of x*sin(7/x) is 7

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limit as x approaches infinity of x*sin(7/x)

x \to \infty lim (x \cdot sin (\frac{7}{x}))

7

Solution steps

x \to \infty lim (x sin (\frac{7}{x}))

Rewrite for L'Hopital

= x \to \infty lim (\frac{sin ( \frac{7}{x} )}{\frac{1}{x}})

Apply L'Hopital's Rule

: x \to \infty lim \frac{- \frac{7 c o s ( \frac{7}{x} )}{x ^{2}}}{- \frac{1}{x ^{2}}}

= x \to \infty lim \frac{- \frac{7 c o s ( \frac{7}{x} )}{x ^{2}}}{- \frac{1}{x ^{2}}}

Simplify

\frac{- \frac{7 c o s ( \frac{7}{x} )}{x ^{2}}}{- \frac{1}{x ^{2}}} : 7 cos (\frac{7}{x})

= x \to \infty lim (7 cos (\frac{7}{x}))

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

= 7 \cdot x \to \infty lim (cos (\frac{7}{x}))

Apply the Limit Chain Rule:

1

= 7 \cdot 1

Simplify

= 7

x \to 1 lim (\frac{x ^{2} - 3}{x - 9})

x \to - 1 + lim (1)

x \to 0 lim (x^{2} - x - 3)

x \to 0 lim ((e)^{\frac{1}{x}})

x \to \frac{π}{2} lim (- 5 sin (x))

What is the limit as x approaches infinity of x*sin(7/x) ?
The limit as x approaches infinity of x*sin(7/x) is 7