Advertisements
Advertisements
Question
Evaluate the following: `lim_(x -> 0) [(2^x - 1)^2/((3^x - 1) xx log (1 + x))]`
Advertisements
Solution
`lim_(x -> 0) [(2^x - 1)^2/((3^x - 1) xx log (1 + x))]`
= `lim_(x -> 0) ((2^x - 1)^2/x^2)/((3^x - 1*log 1 + x)/x^2) ...[("Divide Numerator and"),("Denominator by" x^2),("As" x -> 0"," x ≠ 0),(therefore x^2 ≠ 0)]`
= `(lim_(x -> 0) ((2^x - 1)/x)^2)/(lim_(x -> 0) [((3^x - 1)/x) xx (log 1 + x)/x]`
= `(lim_(x -> 0) ((2^x - 1)/x)^2)/(lim_(x -> 0) ((3^x - 1)/x) xx lim_(x -> 0) (log 1 + x)/x)`
= `(log 2)^2/(log 3 xx 1) ...[(lim_(x -> 0) ("a"^x - 1)/x = log"a"","),(lim_(x -> 0) (log(1 + x))/x = 1)]`
= `(log 2)^2/log3`
APPEARS IN
RELATED QUESTIONS
Evaluate the following: `lim_(x -> 0)[(log(2 + x) - log( 2 - x))/x]`
Evaluate the following: `lim_(x -> 0)[(log(3 - x) - log(3 + x))/x]`
Evaluate the following: `lim_(x -> 2) [(3^(x/2) - 3)/(3^x - 9)]`
Evaluate the following Limits: `lim_(x -> 0)[(5^x - 1)/x]`
Evaluate the following Limits: `lim_(x -> 0)[(log(1 + 9x))/x]`
Evaluate the following Limits: `lim_(x -> 0) ("e"^x + e^(-x) - 2)/x^2`
Evaluate the following Limits: `lim_(x -> 0)[((5^x - 1)^2)/(x*log(1 + x))]`
Evaluate the following Limits: `lim_(x -> 0)[(log(4 - x) - log(4 + x))/x]`
Evaluate the following limit :
`lim_(x -> 0) [(5^x + 3^x - 2^x - 1)/x]`
Evaluate the following limit :
`lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
Evaluate the following limit :
`lim_(x -> 0)[(15^x - 5^x - 3^x + 1)/(x*sinx)]`
Evaluate the following :
`lim_(x -> 0)[("e"^x + "e"^-x - 2)/(x*tanx)]`
`lim_(x -> 0) (log(1 + (5x)/2))/x` is equal to ______.
`lim_(x -> 0) (sin^4 3x)/x^4` = ________.
Evaluate the following `lim_(x->0)[((25)^x - 2(5)^x+1) /(x^2)]`
Evaluate the following:
`lim_(x->0)[((25)^x - 2(5)^x + 1)/x^2]`
Evaluate the following:
`lim_(x -> 0)[((25)^x - 2(5)^x + 1)/x^2]`
Evaluate the limit:
`lim_(z->2)[(z^2-5x+6)/(z^2-4)]`
Evaluate the following:
`lim_(x->0)[((25)^x - 2(5)^x + 1)/x^2]`
