What is the limit as x approaches pi/4 of csc(x) ?

The limit as x approaches pi/4 of csc(x) is square root of 2

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limit as x approaches pi/4 of csc(x)

x \to \frac{π}{4} lim (csc (x))

2

Decimal

1.41421 \dots

Solution steps

x \to \frac{π}{4} lim (csc (x))

Plug in the value

x = \frac{π}{4}

= csc (\frac{π}{4})

Use the following trivial identity:

csc (\frac{π}{4}) = 2

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= 2

What is the limit as x approaches pi/4 of csc(x) ?
The limit as x approaches pi/4 of csc(x) is square root of 2