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

The limit as x approaches 0 of 2xcsc(x) is 2

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limit as x approaches 0 of 2xcsc(x)

x \to 0 lim (2 x csc (x))

2

Solution steps

x \to 0 lim (2 x csc (x))

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

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= 2

x \to 1 lim (\frac{x ^{2} - 4}{x + 2})

x \to 0 lim (ln (sin (∣ x ∣)))

x \to - 2 lim (2 \frac{x}{x ^{2} - 4})

x \to - \infty lim (e^{x} - 2)

x \to - 8 - lim (\frac{3 - 4 x}{x + 8})

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