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