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Question
Evaluate the following limits: `lim_(z -> 4) [(3 - sqrt(5 + z))/(1 - sqrt(5 - z))]`
Solution
`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))/(1 + sqrt(5 - z)) xx (1 + sqrt(5 - z))/(3 + sqrt(5 + z))]`
= `lim_(z -> 4)[(9 - (5 + z))/(1 - (5 - z)) xx (1 + sqrt(5 - z))/(3 + sqrt(5 + z))]`
= `lim_(z -> 4)[(4 - z)/(-4 + z) xx (1 + sqrt(5 - z))/(3 + sqrt(5 + z))]`
= `lim_(z -> 4)[(-(z - 4))/(z - 4) xx (1 + sqrt(5 - z))/(3 + sqrt(5 + z))]`
= `lim_(z -> 4) [(-1 + sqrt(5 - z))/(3 + sqrt(5 + z))] ...[(because z -> 4"," therefore z ≠ 4","),(therefore z - 4 ≠0)]`
= `(-(1 + sqrt(5 - 4)))/(3 + sqrt(5 + 4)`
= `(-(1 + 1))/(3 + 3)`
= `(-2)/6`
= `-1/3`
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