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Question
Evaluate the following limits:
`lim_(x -> 0) (sinalphax)/(sinbetax)`
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Solution
We know `lim_(x -> 0) sinx/x` = 1
`lim_(x -> 0) (sin alpha x)/(sin betax) = lim_(x -> 0) (sin alphax)/(1/alpha (alphax)) xx (1/beta (betax))/(sin betax)`
= `alpha/beta lim_(x -> 0) (sin(alphax))/((alphax)) xx (betax)/(sin(betax))`
= `alpha/beta lim_(alphax -> 0) (sin(alphax))/(alphax) xx lim_(betax -> 0) (betax)/(sin(betax))`
= `alpha/beta lim_(alphax -> 0) (sin(alphax))/(alphax) xx 1/(lim_(betax -> 0) (sin("betax))/(betax))`
= `alpha/beta xx 1 xx 1/1`
`lim_(x -> 0) (sinalphax)/(sinbetax) = alpha/beta`
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