Question

Evaluate the following :

`lim_(x -> "a") [(sinx - sin"a")/(x - "a")]`

Answer 1

`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