What is the limit as x approaches 0 of (2x)/(tan(x)) ?

The limit as x approaches 0 of (2x)/(tan(x)) is 2

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limit as x approaches 0 of (2x)/(tan(x))

x \to 0 lim (\frac{2 x}{tan ( x )})

2

Solution steps

x \to 0 lim (\frac{2 x}{tan ( x )})

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

= 2 \cdot x \to 0 lim (\frac{x}{tan ( x )})

Use the basic trigonometric identity:

\frac{1}{tan ( x )} = cot (x)

= 2 \cdot x \to 0 lim (x cot (x))

Rewrite for L'Hopital

= 2 \cdot x \to 0 lim (\frac{x}{\frac{1}{c o t ( x )}})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= 2

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What is the limit as x approaches 0 of (2x)/(tan(x)) ?
The limit as x approaches 0 of (2x)/(tan(x)) is 2