What is the limit as x approaches 0 of (5x)/(3sin(x)cos(x)) ?

The limit as x approaches 0 of (5x)/(3sin(x)cos(x)) is 5/3

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limit as x approaches 0 of (5x)/(3sin(x)cos(x))

x \to 0 lim (\frac{5 x}{3 sin ( x ) cos ( x )})

\frac{5}{3}

Decimal Notation

1.66666 \dots

Solution steps

x \to 0 lim (\frac{5 x}{3 sin ( x ) cos ( x )})

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

= \frac{5}{3} \cdot x \to 0 lim (\frac{x}{sin ( x ) cos ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( 2 x )})

= \frac{5}{3} \cdot x \to 0 lim (\frac{1}{cos ( 2 x )})

Plug in the value

x = 0

= \frac{5}{3} \cdot \frac{1}{cos ( 2 \cdot 0 )}

Simplify

\frac{5}{3} \cdot \frac{1}{cos ( 2 \cdot 0 )} : \frac{5}{3}

= \frac{5}{3}

x \to - \frac{π}{2} lim (cos (x))

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

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

x \to 1 lim ((x - 1)^{x})

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

What is the limit as x approaches 0 of (5x)/(3sin(x)cos(x)) ?
The limit as x approaches 0 of (5x)/(3sin(x)cos(x)) is 5/3