lim θ→0 sin 6θ θ + tan 8θ the limit could be rewrite like 6lim θ→0 (sin 6θ/θ) times lim θ →0 ( θ/θ + tan 8θ). I know 6lim θ→0 (sin 6θ/θ) equals 6, but how to solve lim θ →0 ( θ/θ + tan 8θ)?
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You should use remarcable limits such that:
`lim_(theta->0) (sin theta)/(theta) = 1`
`lim_(theta->0) (tan theta)/(theta) = 1`
You need to form remarcable limits such that:
`lim_(theta->0) ((sin(6theta))*theta + tan(8 theta)) = lim_(theta->0) ((sin (6 theta))/(6 theta))*(6 theta)* theta + lim_(theta->0) ` `(tan(8 theta))/(8 theta)*(8 theta)`
` lim_(theta->0) ((sin(6theta))*theta + tan(8 theta)) = lim_(theta->0) ((sin (6 theta))/(6 theta))*lim_(theta->0) 6 theta^2 + lim_(theta->0) (tan(8 theta))/(8 theta)*lim_(theta->0) 8 theta`
Substituting 0 for `theta` yields:
`lim_(theta->0) ((sin(6theta))*theta + tan(8 theta)) = 1*6*0^2 + 1*8*0`
`lim_(theta->0) ((sin(6theta))*theta + tan(8 theta)) = 0`
Hence, evaluating the given limit, under the given conditions, yields `lim_(theta->0) ((sin(6theta))*theta + tan(8 theta)) = 0.`
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