What is the limit as x approaches 0 of (sin(5x))/(tan(x)) ?

The limit as x approaches 0 of (sin(5x))/(tan(x)) is 5

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limit as x approaches 0 of (sin(5x))/(tan(x))

x \to 0 lim (\frac{sin ( 5 x )}{tan ( x )})

5

Solution steps

x \to 0 lim (\frac{sin ( 5 x )}{tan ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{cos ( 5 x ) \cdot 5}{sec ^{2} ( x )})

= x \to 0 lim (\frac{cos ( 5 x ) \cdot 5}{sec ^{2} ( x )})

Plug in the value

x = 0

= \frac{cos ( 5 \cdot 0 ) \cdot 5}{sec ^{2} ( 0 )}

Simplify

\frac{cos ( 5 \cdot 0 ) \cdot 5}{sec ^{2} ( 0 )} : 5

= 5

x \to 0 - lim (x cot (π x))

x \to 0 lim (\frac{x sin ( x )}{x})

x \to + 0 lim (x^{2} ln (x))

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

x \to 3 lim (\frac{3 - x}{x - 3})

What is the limit as x approaches 0 of (sin(5x))/(tan(x)) ?
The limit as x approaches 0 of (sin(5x))/(tan(x)) is 5