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Question
Evaluate the following limits:
`lim_(x -> 0) (sin("a" + x) - sin("a" - x))/x`
Solution
We know `lim_(x -> 0) (sinx)/x` = 1
Sin C – sin D = `2 cos ("C" + "D")/2 * sin ("C" - "D")/2`
`sin("a" + x) - sin("a" - x) = 2 cos(("a" + x + "a" - x)/2) xx sin(("a" + x ("a" - x))/2)`
= `2 cos ((2"a")/2) sin (("a" + x - "a" + x)/2)`
= `2 cos "a" * sin ((2x)/2)`
= 2 cos a sin x
`lim_(x -> 0) (sin("a" + x) - sin("a" - x))/x = lim_(x -> 0) (2 cos "a" sin x)/x`
= `2cos "a" lim_(x -> 0) (sinx)/x`
= `2 cos "a" xx 1`
= 2 cos a
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