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

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

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

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

0

Solution steps

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

Multiply by the conjugate of

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

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

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

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

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 _{z \to 0} ( 1 )}{lim _{z \to 0} ( cot ( z ) + csc ( z ) )}

z \to 0 lim (1) = 1

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

= - \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

x \to \infty lim (\frac{2}{3} x)

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

x \to 10 lim ((3 x) (x - 12))

x \to 0 + lim (\frac{x}{e ^{- \frac{3}{x}}})

x \to 1 lim (\frac{[ x ] + 2}{3})

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