What is the limit as x approaches 0 of 1/(3xcsc(x)) ?

The limit as x approaches 0 of 1/(3xcsc(x)) is 1/3

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limit as x approaches 0 of 1/(3xcsc(x))

x \to 0 lim (\frac{1}{3 x csc ( x )})

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

x \to 0 lim (\frac{1}{3 x csc ( x )})

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

= \frac{1}{3} \cdot x \to 0 lim (\frac{1}{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{1}{3} \cdot \frac{lim _{x \to 0} ( 1 )}{lim _{x \to 0} ( x csc ( x ) )}

x \to 0 lim (1) = 1

x \to 0 lim (x csc (x)) = 1

= \frac{1}{3} \cdot \frac{1}{1}

Simplify

= \frac{1}{3}

x \to 0 + lim (- 1 + ln (x) (x))

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

x \to - 5 lim (\frac{x + 5}{x})

x \to 3 lim (\frac{0}{0})

x \to 0 lim (\frac{x}{x ^{2} - 7 x})

What is the limit as x approaches 0 of 1/(3xcsc(x)) ?
The limit as x approaches 0 of 1/(3xcsc(x)) is 1/3