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

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

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

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

1

Solution steps

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

Rewrite for L'Hopital

= x \to 0 lim (\frac{sin ( sin ( x ) )}{\frac{1}{c s c ( x )}})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{cos ( sin ( 0 ) ) cos ( 0 )}{cos ( 0 )}

Simplify

\frac{cos ( sin ( 0 ) ) cos ( 0 )}{cos ( 0 )} : 1

= 1

t \to \infty lim (\frac{ln ( 3 t )}{t ^{2}})

x \to 0 lim (2 x + 3)

x \to 1 lim (x^{2} + 2 x + 5)

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

x \to - 5 lim (- 35 + 3 x + 2 x^{2})

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