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Question
Evaluate the following limits:
`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) ("a" > "b")`
Solution
`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2), "a" > "b"`
`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) = lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) xx (sqrt(x - "b") + sqrt("a" - "b"))/(sqrt(x - "b") + sqrt("a" - "b"))`
= `lim_(x -> "a") ((x - "b") - ("a"- "b"))/((x^2 - "a"^2) [sqrt(x - "b") + sqrt("a" - "b")]`
= `lim_(x -> "a") (x - "b" - "a" + "b")/((x - "a")(x + "a") [sqrt(x - "b") + sqrt("a" - "b")]`
= `lim_(x -> "a") (x - "a")/((x - "a")(x + "a") [sqrt(x - "b") + sqrt("a" - "b")]`
= `lim_(x -> "a") 1/((x + "a")[sqrt("x" - "b") + sqrt('a" -"b")]`
= `1/(("a" + "a")[sqrt("a" - "b") + sqrt("a" - "b")]`
= `1/(2"a" xx 2sqrt("a" - "b")`
= `1/(4"a"sqrt("a" - "b")`
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