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

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

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

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

1

Solution steps

x \to 0 lim (sin (x) cot (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 1 lim (\frac{1}{0})

x \to \frac{3 π}{4} lim (sin (x))

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

x \to 0 lim ((1 - 9 x)^{\frac{5}{x}})

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

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