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

The limit as x approaches 0 of tan(x)csc(x) is 1

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limit as x approaches 0 of tan(x)csc(x)

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

1

Solution steps

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

Simplify

tan (x) csc (x) : sec (x)

= x \to 0 lim (sec (x))

Plug in the value

x = 0

= sec (0)

Use the following trivial identity:

sec (0) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 1

What is the limit as x approaches 0 of tan(x)csc(x) ?
The limit as x approaches 0 of tan(x)csc(x) is 1