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

The limit as x approaches 0 of xcsc(x)sec(x) is 1

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limit as x approaches 0 of xcsc(x)sec(x)

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

1

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{x sec ( x ) tan ( x ) + sec ( x )}{cos ( x )})

= x \to 0 lim (\frac{x sec ( x ) tan ( x ) + sec ( x )}{cos ( x )})

Plug in the value

x = 0

= \frac{0 \cdot sec ( 0 ) tan ( 0 ) + sec ( 0 )}{cos ( 0 )}

Simplify

\frac{0 \cdot sec ( 0 ) tan ( 0 ) + sec ( 0 )}{cos ( 0 )} : 1

= 1

x \to 0 lim ((2 + x)^{3} - 8 x)

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

t \to 0 lim (\frac{e ^{- 7 t} - 1}{t})

x \to 0 lim (3 - 3 x + 6 x^{2})

x \to - 4 lim (\frac{2}{x ^{2} + 3 x - 4} - \frac{3}{x + 4})

What is the limit as x approaches 0 of xcsc(x)sec(x) ?
The limit as x approaches 0 of xcsc(x)sec(x) is 1