What is the limit as x approaches 0-of 4(1/x-cot(x)) ?

The limit as x approaches 0-of 4(1/x-cot(x)) is 0

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limit as x approaches 0-of 4(1/x-cot(x))

x \to 0 - lim (4 (\frac{1}{x} - cot (x)))

0

Solution steps

x \to 0 - lim (4 (\frac{1}{x} - cot (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} - cot (x))

Multiply by the conjugate of

\frac{1}{x} - cot (x) : \frac{\frac{1}{x ^{2}} - cot ^{2} ( x )}{\frac{1}{x} + cot ( x )}

= 4 \cdot x \to 0 - lim (\frac{\frac{1}{x ^{2}} - cot ^{2} ( x )}{\frac{1}{x} + cot ( 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 -} ( \frac{1}{x ^{2}} - cot ^{2} ( x ) )}{lim _{x \to 0 -} ( \frac{1}{x} + cot ( x ) )}

x \to 0 - lim (\frac{1}{x ^{2}} - cot^{2} (x)) = \frac{2}{3}

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

= 4 \cdot \frac{\frac{2}{3}}{- \infty}

Simplify

4 \cdot \frac{\frac{2}{3}}{- \infty} : 0

= 0

x \to - 43 lim (x^{2} - 4 x + 8)

x \to 2 lim (3 x^{2} + 2 x - 10)

x \to 1 lim (arctan (\frac{1}{x}))

x \to 0 lim ((1 + \frac{x}{3})^{2})

x \to 1 lim (\frac{1 - cos ( 2 x )}{x})

What is the limit as x approaches 0-of 4(1/x-cot(x)) ?
The limit as x approaches 0-of 4(1/x-cot(x)) is 0