Advertisements
Advertisements
Question
Evaluate the following: `lim_(x -> 0) [("a"^(3x) - "b"^(2x))/(log 1 + 4x)]`
Advertisements
Solution
`lim_(x -> 0) [("a"^(3x) - "b"^(2x))/(log 1 + 4x)]`
= `lim_(x -> 0) ("a"^(3x) - 1 - "b"^(2x) - 1)/(log 1 + 4x)`
= `lim_(x -> 0) [("a"^(3x) - 1 - "b"^(2x) - 1)/x)/((log 1 + 4x)/x)`
= `(lim_(x -> 0) [("a"^(3x) - 1)/x - ("b"^(2x) - 1)/x])/(lim_(x -> 0) (log 1 + 4x)/(x)`
= `(lim_(x -> 0)[("a"^(3x) - 1)/(3x)] xx 3 - lim_(x -> 0)[("b"^(2x) - 1)/(2x)] xx 2)/(lim_(x -> 0) (log 1 + 4x)/(4x) xx 4)`
= `(3log "a" - 2 log"b")/(1 xx 4) ...[(because x -> 0"," 2x -> 0"," 3x -> 0),(4x -> 0 and lim_(x -> 0) ("a"^x - 1)/x = log "a"),(and lim_(x -> 0) (log (1 + x))/x = 1)]`
= `1/4(log"a"^3 - log"b"^2)`
= `1/4 log("a"^3/"b"^2)`
APPEARS IN
RELATED QUESTIONS
Evaluate the following: `lim_(x -> 0) [(3^x + 3^-x - 2)/x^2]`
Evaluate the following Limits: `lim_(x -> 0)[(log(1 + 9x))/x]`
Evaluate the following Limits: `lim_(x -> 0)((1 - x)^5 - 1)/((1 - x)^3 - 1)`
Evaluate the following Limits: `lim_(x -> 0)[("a"^x + "b"^x + "c"^x - 3)/x]`
Evaluate the following Limits: `lim_(x -> 0) ("e"^x + e^(-x) - 2)/x^2`
Evaluate the following Limits: `lim_(x -> 0)[(x(6^x - 3^x))/((2^x - 1)*log(1 + x))]`
Evaluate the following Limits: `lim_(x -> 0)[(log(4 - x) - log(4 + x))/x]`
Evaluate the following limit :
`lim_(x -> 0) [(9^x - 5^x)/(4^x - 1)]`
Evaluate the following limit :
`lim_(x -> 0) [(3^x + 3^-x - 2)/(x*tanx)]`
Evaluate the following limit :
`lim_(x -> 0) [((49)^x - 2(35)^x + (25)^x)/(sinx* log(1 + 2x))]`
Select the correct answer from the given alternatives.
`lim_(x -> 0) [(log(5 + x) - log(5 - x))/sinx]` =
Evaluate the following :
`lim_(x -> 0)[("e"^x + "e"^-x - 2)/(x*tanx)]`
Evaluate the following :
`lim_(x -> 1) [("ab"^x - "a"^x"b")/(x^2 - 1)]`
Evaluate the following :
`lim_(x -> 0) [((5^x - 1)^2)/((2^x - 1)log(1 + x))]`
The value of `lim_{x→0} (1 + sinx - cosx + log_e(1 - x))/x^3` is ______
Evaluate the following:
`lim_(x->0)[((25)^x - 2(5)^x + 1)/x^2]`
Evaluate the following limit :
`lim(x>2)[(z^2 -5z+6)/(z^2-4)]`
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]`
