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Question
Evaluate the following :
`lim_(x -> "a") [(x cos "a" - "a" cos x)/(x - "a")]`
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Solution
`lim_(x -> "a") [(x cos "a" - "a" cos x)/(x - "a")]`
= `lim_(x -> "a") [(x cos "a" - "a" cos "a" + "a" cos "a" - "a" cos x)/(x - "a")]` ...[Note this step]
= `lim_(x -> "a") [((x - "a") cos "a" + "a"(cos"a" - cosx))/(x - "a")]`
= `lim_(x -> "a") [((x - "a") cos "a" + 2"a" sin (("a" + x)/2)((x - "a")/2))/(x - "a")]`
= `lim_(x -> "a") [((x - "a")cos"a")/(x - "a") + (2"a" sin (("a" + x)/2) sin((x - "a")/2))/(x - "a")]`
= `lim_(x -> "a") [cos"a" + "a" sin (("a" + x)/2)* (sin((x - "a")/2))/(((x - "a")/2))]` ...[∵ x → a, x ≠ a, ∴ x – a ≠ 0]
= `lim_(x -> "a") cos"a" + "a"[lim_(x -> "a") sin(("a" + x)/2)] xx [lim_(x -> "a") sin((x - "a")/2)/((x - "a")/2)]`
= `cos "a" + "a" sin (("a" + "a")/2) xx 1 ...[(because x -> "a" "," x ≠ "a" therefore (x - "a")/2 -> 0),(and lim_(theta -> 0) sintheta/theta = 1)]`
= cos a + a sin a
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