What is the limit as x approaches 0 of 12xcot(9x) ?

The limit as x approaches 0 of 12xcot(9x) is 4/3

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limit as x approaches 0 of 12xcot(9x)

x \to 0 lim (12 x cot (9 x))

\frac{4}{3}

Decimal Notation

1.33333 \dots

Solution steps

x \to 0 lim (12 x cot (9 x))

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

= 12 \cdot x \to 0 lim (x cot (9 x))

Rewrite for L'Hopital

= 12 \cdot x \to 0 lim (\frac{x}{\frac{1}{c o t ( 9 x )}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{sec ^{2} ( 9 x ) \cdot 9})

= 12 \cdot x \to 0 lim (\frac{1}{sec ^{2} ( 9 x ) \cdot 9})

Plug in the value

x = 0

= 12 \cdot \frac{1}{sec ^{2} ( 9 \cdot 0 ) \cdot 9}

Simplify

12 \cdot \frac{1}{sec ^{2} ( 9 \cdot 0 ) \cdot 9} : \frac{4}{3}

= \frac{4}{3}

x \to \infty lim (\frac{ln ( 1 + e ^{x} )}{12 x + 11})

x \to 1 lim (x^{2} + 6 x)

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

x \to 2 lim (3 x^{2} + 7 x)

x \to 2 + lim (x + 1)

What is the limit as x approaches 0 of 12xcot(9x) ?
The limit as x approaches 0 of 12xcot(9x) is 4/3