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

The limit as x approaches 0 of cot(x)-csc(x) is 0

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limit as x approaches 0 of cot(x)-csc(x)

x \to 0 lim (cot (x) - csc (x))

0

Solution steps

x \to 0 lim (cot (x) - csc (x))

Multiply by the conjugate of

cot (x) - csc (x) : \frac{- 1}{cot ( x ) + csc ( x )}

= x \to 0 lim (\frac{- 1}{cot ( x ) + csc ( x )})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= - x \to 0 lim (\frac{1}{cot ( x ) + csc ( x )})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= - \frac{lim _{x \to 0} ( 1 )}{lim _{x \to 0} ( cot ( x ) + csc ( x ) )}

x \to 0 lim (1) = 1

x \to 0 lim (cot (x) + csc (x)) = \infty

= - \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

What is the limit as x approaches 0 of cot(x)-csc(x) ?
The limit as x approaches 0 of cot(x)-csc(x) is 0