What is the limit as x approaches 0 of xcsc(x)cos(x) ?

The limit as x approaches 0 of xcsc(x)cos(x) is 1

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limit as x approaches 0 of xcsc(x)cos(x)

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

1

Solution steps

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

Simplify

x csc (x) cos (x) : x cot (x)

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

Rewrite for L'Hopital

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( x )}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{1}{sec ^{2} ( x )})

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 0 )}

Simplify

\frac{1}{sec ^{2} ( 0 )} : 1

= 1

What is the limit as x approaches 0 of xcsc(x)cos(x) ?
The limit as x approaches 0 of xcsc(x)cos(x) is 1