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Question
Evaluate the following limit.
`lim_(x -> 0) (sin ax)/(sin bx), a, b != 0`
Solution
`lim_(x -> 0) (sin ax)/(sin bx), a, b != 0`
At x = 0, the value of the given function takes the form `0/0`.
Now, `lim_(x → 0) (sinax)/(sin bx)` = `lim_(x → 0) (((sinax)/(ax)) xx ax)/(((sinbx)/(bx)) xx bx)`
= `(a/b) xx (lim_(ax → 0) ((sinax)/(ax)))/(lim_(bx → 0) ((sinbx)/(bx)))` `[("x"-> 0 =>ax ->0),("and" x-> 0 => bx -> 0)]`
= `(a/b) xx 1/1` `[lim_(y->0) (siny)/y = 1)]`
= `a/b`
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