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Question
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/(sin 5x)`
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Solution
We know `lim_(x -> 0) (sin x)/x` = 1
`lim_(x -> 0) (tan 2x)/(sin 5x) = lim_(x -> 0) (sin 2x)/(cos 2x) xx 1/(sin 5x)`
= `lim_(x -> 0) (sin 2x)/(1/2 (2x)) xx 1/(cos 2x) xx (1/5(5x))/(sin 5x)`
= `2/5 (lim_(2x-> 0) (sin 2x)/(2x)) (lim_(x -> 0) 1/(cos 2x)) xx (1/(lim_(5x -> 0) (sin 5x)/(5x)))`
= `2/5 xx 1 xx 1/cos 0 xx 1/1`
`lim_(x -> 0) (tan 2x)/(sin 5x) = 2/5 xx 1/1 xx 1`
= `2/5`
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