Advertisements
Advertisements
Question
Evaluate the following limit :
`lim_(x -> 0)[(sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4))]`
Solution
`lim_(x -> 0)(sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4))`
= `lim_(x -> 0) (sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4)) xx (sqrt(x^2 + 9) + sqrt(2x^2 + 9))/(sqrt(x^2 + 9) + sqrt(2x^2 + 9)) xx (sqrt(3x^2 + 4) + sqrt(2x^2 + 4))/(sqrt(3x^2 + 4) + sqrt(2x^2 + 4))`
= `lim_(x -> 0) ([(x^2 + 9) - (2x^2 + 9)][sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/([(3x^2 + 4) - (2x^2 + 4)][sqrt(x^2 + 9) + sqrt(2x^2 + 9)])`
= `lim_(x -> 0) (-x^2[sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/(x^2[sqrt(x^2 + 9) + sqrt(2x^2 + 9)]`
= `lim_(x -> 0) (-[sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/(sqrt(x^2 + 9) + sqrt(2x^2 + 9)) ...[(because x -> 0"," x ≠ 0),(therefore x^2 ≠ 0)]`
= `(-lim_(x -> 0) [sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/(lim_(x -> 0) [sqrt(x^2 + 9) + sqrt(2x^2 + 9)])`
= `(-[sqrt(0 + 4) + sqrt(0 + 4)])/(sqrt(0 + 9) + sqrt(0 + 9)`
= `- ((2 + 2))/(3 + 3)`
= `-2/3`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limits: `lim_(x -> 0) [(sqrt(6 + x + x^2) - sqrt(6))/x]`
Evaluate the following limits: `lim_(x -> 2)[(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2))]`
Evaluate the following limits: `lim_(x -> "a") [(sqrt("a" + 2x) - sqrt(3x))/(sqrt(3"a" + x) - 2sqrt(x))]`
Evaluate the following limits: `lim_(x -> 2)[(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2))]`
Evaluate the following limits: `lim_(x -> 0)[(sqrt(1 + x^2) - sqrt(1 + x))/(sqrt(1 + x^3) - sqrt(1 + x))]`
``Evaluate the following limits: `lim_(x -> 4) [(x^2 + x - 20)/(sqrt(3x + 4) - 4)]`
Evaluate the following limits: `lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))]`
Evaluate the following limits: `lim_(y -> 2) [(2 - y)/(sqrt(3 - y) - 1)]`
Evaluate the following limits: `lim_(z -> 4) [(3 - sqrt(5 + z))/(1 - sqrt(5 - z))]`
Evaluate the following limit :
`lim_(x -> 2) [(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2))]`
Evaluate the following limit :
`lim_(x -> "a") [(sqrt("a" + 2x) - sqrt(3x))/(sqrt(3"a" + x) - 2sqrt(x))]`
Evaluate the following limit :
`lim_(x -> 2) [(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2))]`
Evaluate the following limit :
`lim_(x -> 2)[(sqrt(1 + sqrt(2 + x)) - sqrt(3))/(x - 2)]`
Evaluate the following limit :
`lim_(y -> 0) [(sqrt("a" + y) - sqrt("a"))/(ysqrt("a" + y))]`
Evaluate the following limit :
`lim_(x -> 1) [(x^2 + xsqrt(x) - 2)/(x - 1)]`
Evaluate the Following limit :
`lim_(x -> 0) [(sqrt(1 + x^2) - sqrt(1 + x))/(sqrt(1 + x^3) - sqrt(1 - x))]`
Evaluate the Following limit :
`lim_(x -> 4) [(x^2 + x - 20)/(sqrt(3x + 4) - 4)]`
Evaluate the Following limit :
`lim_(z -> 4) [(3 - sqrt(5 + z))/(1 - sqrt(5 - z))]`
Evaluate the Following limit :
`lim_(x -> 0)[3/(xsqrt(9 - x)) - 1/x]`
Evaluate the following :
`lim_(x -> 0)[x]` ([*] is a greatest integer function.)
Evaluate the following limit:
`lim_(x->0)[(sqrt(6 + x + x^2) - sqrt6)/x]`
Evaluate the following limit:
`lim_(x->0)[(sqrt(6+x+x^2)-sqrt6)/x]`
Evaluate the following limit:
`lim_(x->0)[(sqrt(6+x+x^2)-sqrt6)/x]`
Evaluate the following limit.
`lim_(x→0) [[sqrt(6 + x + x^2)- sqrt6]/x]`
Evaluate the following limit:
`lim_(x->0)[(sqrt(6 + x + x^2) - sqrt6)/ (x)]`
Evaluate the following limit:
`lim_(x -> 0)[(sqrt(6 + x + x^2) - sqrt6)/x]`
