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Question
Evaluate the following :
`lim_(x -> "a") [(sinx - sin"a")/(x - "a")]`
Solution
`lim_(x -> "a") (sinx - sin"a")/(x - "a")`
Put x = a + h,
∴ x – a = h
As x → a, h → 0
∴ `lim_(x -> "a") (sinx - sin"a")/(x - "a")`
= `lim_("h" -> 0) (sin "a" + "h" - sin"a")/"h"`
= `lim_("h" -> 0) (2cos (("a" + "h" + "a")/2) sin(("a" + "h" - "a")/2))/"h"`
= `lim_("h" -> 0) (2cos("a" + "h"/2) sin "h"/2)/"h"`
= `lim_("h" -> 0) cos ("a" + "h"/2) * lim_("h" -> 0) (sin("h"/2))/(("h"/2))`
= `cos ("a" + 0)(1) ...[because "h" -> 0, "h"/2 -> 0 "and" lim_(theta -> 0) sintheta/theta = 1]`
= cos a
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