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Evaluate: lim(theta rarr 0) "cot" theta...

Evaluate: `lim_(theta rarr 0) "cot" theta - "cosec" theta`.

A

0

B

2

C

4

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(theta rarr 0)("lim") "cot" theta - "cosec" theta = oo - oo`, i.e., indeterminate form.
`"cot" theta - "cosec" theta = ("cos" theta)/("sin" theta)-(1)/("sin" theta)=("cos" theta -1)/("sin" theta)`
`("cos"^(2) theta -1)/("sin" theta("cos" theta+1)`
`(-"sin" theta)/("sin" theta("cos" theta+1))=(-"sin" theta)/("cos" theta+1)`
`therefore underset(theta rarr 0)("lim")"cot" theta - "cosec" theta`
`= underset(theta rarr 0)("lim")(-"sin" theta)/("cos" theta+1)=(-"sin 0")/("cos" 0+1)=(-0)/(1+1)=0`.
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