What is the limit as x approaches pi+of 2csc(2x) ?

The limit as x approaches pi+of 2csc(2x) is infinity

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limit as x approaches pi+of 2csc(2x)

x \to π + lim (2 csc (2 x))

\infty

Solution steps

x \to π + lim (2 csc (2 x))

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

= 2 \cdot x \to π + lim (csc (2 x))

Express with sin, cos

= 2 \cdot x \to π + lim (\frac{1}{sin ( 2 x )})

For

x

approaching

π

from the right

, x > π \Rightarrow sin (2 x) > 0

The denominator is a positive quantity approaching 0 from the right

= 2 \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

x \to \frac{π}{4} lim (\frac{sin ( 2 x )}{sin ( x ) tan ( x )})

x \to \infty lim (\frac{( n + 1 ) ^{n + 1}}{( n + 2 ) ^{n + 2}})

x \to 1 lim (\frac{x ^{2} - 1}{1 - x ^{3}})

x \to 1 lim (cos (2 x))

x \to 2 lim (8 - 3 x)

What is the limit as x approaches pi+of 2csc(2x) ?
The limit as x approaches pi+of 2csc(2x) is infinity