What is the limit as x approaches 0 of (xsin(x))/x ?

The limit as x approaches 0 of (xsin(x))/x is 0

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limit as x approaches 0 of (xsin(x))/x

x \to 0 lim (\frac{x sin ( x )}{x})

0

Solution steps

x \to 0 lim (\frac{x sin ( x )}{x})

Cancel the common factor:

x

= x \to 0 lim (sin (x))

Plug in the value

x = 0

= sin (0)

Use the following trivial identity:

sin (0) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

What is the limit as x approaches 0 of (xsin(x))/x ?
The limit as x approaches 0 of (xsin(x))/x is 0