What is the limit as x approaches 0 of csc(x)ln(x+1) ?

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

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limit as x approaches 0 of csc(x)ln(x+1)

x \to 0 lim (csc (x) ln (x + 1))

1

Solution steps

x \to 0 lim (csc (x) ln (x + 1))

Express with sin, cos

= x \to 0 lim (\frac{ln ( x + 1 )}{sin ( x )})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{\frac{1}{0 + 1}}{cos ( 0 )}

Simplify

\frac{\frac{1}{0 + 1}}{cos ( 0 )} : 1

= 1

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