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Question
Discuss the continuity of function f at x = 0.
Where f(X) = `[ [sqrt ( 4 + x ) - 2 ]/ ( 3x )]`, For x ≠ 0
= `1/12`, For x = 0
Solution
Consider `lim_( x -> 0) f(x)`
= `lim_( x -> 0 ) [[sqrt ( 4 + x ) - 2 ]/ ( 3x )]`
= `lim_( x -> 0 ) [[sqrt ( 4 + x ) - 2 ]/( 3x ) xx [sqrt ( 4 + x ) +2]/ [sqrt ( 4 + x ) + 2]]`
= `lim_( x -> 0) [[ 4 + x - 4 ]/[ 3x ( sqrt( 4 + x ) + 2) ]] `
= `lim_( x -> 0) [[ x ]/[ 3x ( sqrt( 4 + x ) + 2) ]] `
= `lim_( x -> 0) [[ 1 ]/[ 3 ( sqrt( 4 + x ) + 2) ]] `
( ∵ x → 0 x ≠ 0 )
= `[ 1 ]/[ 3 (sqrt 4 + 2)] = 1/12`
Given f(0) = `1/12`
`lim_( x -> 0) f(x) = f(0)`
∴ f is continuous at x = 0.
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