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

The limit as x approaches 0 of (xcos(5x))/(sin(5x)) is 1/5

Popular >

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

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

\frac{1}{5}

Decimal Notation

0.2

Solution steps

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

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

= x \to 0 lim (x cot (5 x))

Rewrite for L'Hopital

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( 5 x )}})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 5 \cdot 0 ) \cdot 5}

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

= \frac{1}{5}

x \to \frac{π}{2} lim (x tan (x))

x \to 0 lim (3 \frac{x}{sin ( x )})

x \to 0 + lim (x^{\frac{10}{x}})

x \to 0 - lim (6 (cot (x) - \frac{1}{x}))

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

What is the limit as x approaches 0 of (xcos(5x))/(sin(5x)) ?
The limit as x approaches 0 of (xcos(5x))/(sin(5x)) is 1/5