Given:
limx→0 (cosec x − cot x)
Step 1: Use identity
cosec x − cot x = (1 − cos x) / sin x
Step 2: Rewrite the limit
limx→0 (1 − cos x) / sin x
Step 3: Multiply numerator and denominator by (1 + cos x)
= limx→0 [(1 − cos x)(1 + cos x)] / [sin x(1 + cos x)]
= limx→0 (1 − cos2 x) / [sin x(1 + cos x)]
= limx→0 sin2 x / [sin x(1 + cos x)]
= limx→0 sin x / (1 + cos x)
Step 4: Apply standard limits
As x → 0,
sin x → 0
cos x → 1
Therefore,
limx→0 sin x / (1 + cos x) = 0 / 2
= 0
Final Answer:
limx→0 (cosec x − cot x) = 0
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