What is the limit as x approaches 0+of tan(5x)ln(8x) ?

The limit as x approaches 0+of tan(5x)ln(8x) is 0

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limit as x approaches 0+of tan(5x)ln(8x)

x \to 0 + lim (tan (5 x) ln (8 x))

0

Solution steps

x \to 0 + lim (tan (5 x) ln (8 x))

Rewrite for L'Hopital

= x \to 0 + lim (\frac{ln ( 8 x )}{\frac{1}{t a n ( 5 x )}})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{1}{x}}{- csc ^{2} ( 5 x ) \cdot 5})

= x \to 0 + lim (\frac{\frac{1}{x}}{- csc ^{2} ( 5 x ) \cdot 5})

Simplify

\frac{\frac{1}{x}}{- csc ^{2} ( 5 x ) \cdot 5} : - \frac{1}{5 x csc ^{2} ( 5 x )}

= x \to 0 + lim (- \frac{1}{5 x csc ^{2} ( 5 x )})

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

= - \frac{1}{5} \cdot x \to 0 + lim (\frac{1}{x csc ^{2} ( 5 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}{5} \cdot \frac{lim _{x \to 0 +} ( 1 )}{lim _{x \to 0 +} ( x csc ^{2} ( 5 x ) )}

x \to 0 + lim (1) = 1

x \to 0 + lim (x csc^{2} (5 x)) = \infty

= - \frac{1}{5} \cdot \frac{1}{\infty}

Simplify

- \frac{1}{5} \cdot \frac{1}{\infty} : 0

= 0

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What is the limit as x approaches 0+of tan(5x)ln(8x) ?
The limit as x approaches 0+of tan(5x)ln(8x) is 0