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प्रश्न
Evaluate the Following limit :
`lim_(z -> 4) [(3 - sqrt(5 + z))/(1 - sqrt(5 - z))]`
उत्तर
`lim_(z -> 4) (3 - sqrt(5 + z))/(1 - sqrt(5 - z))`
= `lim_(z -> 4) (3 - sqrt(5 + z))/(1 - sqrt(5 - z)) xx (3 + sqrt(5 + z))/(3 + sqrt(5 + z)) xx (1 + sqrt(5 - z))/(1 + sqrt(5 - z))`
= `lim_(z -> 4) ([9 - (5 + z)][1 + sqrt(5 - z)])/([1 - (5 - z)][3 + sqrt(5 + z)])`
= `lim_(z -> 4) (-(z - 4)[1 + sqrt(5 - z)])/((z - 4)[3 + sqrt(5 + z)])`
= `lim_(z -> 4) (-[1 + sqrt(5 - z)])/([3 + sqrt(5 + z)]) ...[(because z -> 4"," z ≠ 4),(therefore z - 4 ≠ 0)]`
= `(-lim_(z -> 4) [1 + sqrt(5 - z)])/(lim_(z -> 4) [3 + sqrt(5 + z)])`
= `(-[1 + sqrt(5 - 4)])/([3 + sqrt(5 + 4)])`
= `(-[1 + 1])/(3 + 3)`
= `-1/3.`
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