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Question
Evaluate: `lim_(x -> 0) (sin 2x + 3x)/(2x + tan 3x)`
Solution
Given that `lim_(x -> 0) (sin 2x + 3x)/(2x + tan 3x)`
= `lim_(x -> 0) (((sin 2x + 3x)/(2x)) xx 2x)/(((2x + tan 3x)/(3x)) xx 3x)`
= `lim_(x -> 0) (((sin 2x)/(2x) + (3x)/(2x)) xx 2x)/(((2x)/(3x) + (tan 3x)/(3x)) xx 3x)`
= `((lim_(2x -> 0) (sin 2x)/(2x) + 3/2))/([2/3 + lim_(3x -> 0) (tan 3x)/(3x)]) xx 2/3` .....`[because lim_(x -> 0) sinx/x = 1]`
= `((1 + 3/2)/(2/3 + 1)) xx 2/3` .....`[because lim_(x -> 0) tanx/x = 1]`
= `(5/2)/(5/3) xx 2/3`
= `3/2 xx 2/3`
= 1
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