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Question
Evaluate the following :
`lim_(x -> "a") [(sinx - sin"a")/(root(5)(x) - root(5)("a"))]`
Solution
`lim_(x -> "a") (sinx - sin"a")/(root(5)(x) - root(5)("a"))`
= `lim_(x -> "a") (2cos((x + "a")/2) sin((x - "a")/2))/(root(5)(x) - root(5)("a"))`
= `lim_(x -> "a") (2cos((x + "a")/2) sin((x - "a")/2)/(((x - "a")/2)))/(((root(5)(x) - root(5)("a"))/(x - "a"))) xx 1/2 ...[(because x -> "a""," x ≠ "a"),(therefore x - "a" ≠ 0)]`
= `([lim_(x -> "a") cos((x + "a")/2)] xx [lim_(x -> "a") (sin((x - "a")/2))/(((x - "a")/2))])/(lim_(x -> "a") ((x^(1/5) - "a"^(1/5))/(x - "a"))`
= `(cos(("a" + "a")/2)*1)/(1/5"a"^(-4/5)) ...[(because x -> "a""," x ≠ "a" therefore (x - "a")/2 ->0),("and" lim_(theta -> 0) (sin theta)/theta = 1),("Also""," lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1))]`
= `5"a"^(4/5) cos"a"`
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