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Evaluate the following limit : limz→a[(z+2)32-(a+2)32z-a]

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Question

Evaluate the following limit :

`lim_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`

Sum
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Solution

`lim_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`

= `lim_(z -> "a") (("z" + 2)^(3/2) - ("a" + 2)^(3/2))/((z + 2) - ("a" + 2))`

Put z + 2 = y and a + 2 = b

As z → a, z + 2 → a + 2

i.e. y → b

∴ `lim_(z -> "a") ((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")`

= `lim_(y -> "b") (y^(3/2) - "b"^(3/2))/(y - "b")`

= `3/2* "b"^(1/2)      ...[because  lim_(z -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`

= `3/2 sqrt("a" + 2)`     ...[∵ b = a + 2]

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Chapter 7: Limits - Exercise 7.1 [Page 139]

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