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

The limit as x approaches 0 of (xcos(x))/x is 1

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

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

1

Solution steps

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

Cancel the common factor:

x

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

Plug in the value

x = 0

= cos (0)

Use the following trivial identity:

cos (0) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

\frac{d}{d x} (\frac{3}{x ^{2} + 9})

\frac{\partial}{\partial y} (y e^{- x^{2} + y})

\int \frac{x ^{2} + 3}{x x ^{2} - 9} d x

\frac{\partial}{\partial x} (cos^{6} (x^{8} y^{6}))

y = 2 x - x^{2}, y = 2 x - 4

What is the limit as x approaches 0 of (xcos(x))/x ?
The limit as x approaches 0 of (xcos(x))/x is 1