What is the limit as x approaches 0+of xln(tan(3x)) ?

The limit as x approaches 0+of xln(tan(3x)) is 0

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limit as x approaches 0+of xln(tan(3x))

x \to 0 + lim (x ln (tan (3 x)))

0

Solution steps

x \to 0 + lim (x ln (tan (3 x)))

Rewrite for L'Hopital

= x \to 0 + lim (\frac{ln ( tan ( 3 x ) )}{\frac{1}{x}})

Apply L'Hopital's Rule

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

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

Simplify

= x \to 0 + lim (- 3 x^{2} sec^{2} (3 x) cot (3 x))

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

= - 3 \cdot x \to 0 + lim (x^{2} sec^{2} (3 x) cot (3 x))

Express with sin, cos

= - 3 \cdot x \to 0 + lim (\frac{x ^{2}}{cos ( 3 x ) sin ( 3 x )})

\frac{x ^{2}}{cos ( 3 x ) sin ( 3 x )} = \frac{x ^{2}}{sin ( 3 x )} \cdot \frac{1}{cos ( 3 x )}

= - 3 \cdot x \to 0 + lim (\frac{x ^{2}}{sin ( 3 x )} \cdot \frac{1}{cos ( 3 x )})

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

With the exception of indeterminate form

= - 3 \cdot x \to 0 + lim (\frac{x ^{2}}{sin ( 3 x )}) \cdot x \to 0 + lim (\frac{1}{cos ( 3 x )})

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

x \to 0 + lim (\frac{1}{cos ( 3 x )}) = 1

= - 3 \cdot 0 \cdot 1

Simplify

= 0

- \frac{19}{( 2 x - 5 ) ^{2}}

x \to - 26.8 lim (- 26.8)

\frac{4}{x + 2}, 0

x \to \infty lim (10 \cdot e^{\frac{- x}{2}} + 4)

\frac{d y}{d x} = \frac{4 x}{9 y}

What is the limit as x approaches 0+of xln(tan(3x)) ?
The limit as x approaches 0+of xln(tan(3x)) is 0