What is the limit as x approaches 0 of cot(x)sin(x) ?

The limit as x approaches 0 of cot(x)sin(x) is 1

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limit as x approaches 0 of cot(x)sin(x)

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

1

Solution steps

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

Express with sin, cos

= 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

x \to - 12 lim (x + 12)

x \to 3 lim (2 x^{2} + 3)

t \to 50 lim (\frac{100}{50 - t})

x \to 0 lim (\frac{x + 2}{x + 3})

x \to 7 - lim (\frac{1 - 8 x}{x - 7})

What is the limit as x approaches 0 of cot(x)sin(x) ?
The limit as x approaches 0 of cot(x)sin(x) is 1