What is the limit as x approaches 0 of 4/(xcsc(8x)) ?

The limit as x approaches 0 of 4/(xcsc(8x)) is 32

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limit as x approaches 0 of 4/(xcsc(8x))

x \to 0 lim (\frac{4}{x csc ( 8 x )})

32

Solution steps

x \to 0 lim (\frac{4}{x csc ( 8 x )})

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

= 4 \cdot x \to 0 lim (\frac{1}{x csc ( 8 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

= 4 \cdot \frac{lim _{x \to 0} ( 1 )}{lim _{x \to 0} ( x csc ( 8 x ) )}

x \to 0 lim (1) = 1

x \to 0 lim (x csc (8 x)) = \frac{1}{8}

= 4 \cdot \frac{1}{\frac{1}{8}}

Simplify

= 32

\int (4 e^{x} - \frac{5}{x}) d x

x \to - 1 lim (\frac{\frac{1}{x} + 1}{x + 1})

\int_{- 2}^{2} (4 - x^{2})^{2} d x

\int_{- 1}^{1} \frac{1}{x + 2} d x

\int_{1}^{2} \frac{x + 1}{x} d x

What is the limit as x approaches 0 of 4/(xcsc(8x)) ?
The limit as x approaches 0 of 4/(xcsc(8x)) is 32