Advertisements
Advertisements
Question
Evaluate the following Limits: `lim_(x -> 0)[("a"^(3x) - "a"^(2x) - "a"^x + 1)/x^2]`
Advertisements
Solution
`lim_(x -> 0)("a"^(3x) - "a"^(2x) - "a"^x + 1)/x^2`
= `lim_(x -> 0) ("a"^(2x)*"a"^x - "a"^(2x) - "a"^x + 1)/x^2`
= `lim_(x -> 0)("a"^(2x)("a"^x - 1) - 1("a"^x - 1))/x^2`
= `lim_(x -> 0) (("a"^x - 1)*("a"^(2x) - 1))/x^2`
= `lim_(x -> 0) (("a"^x - 1)/x xx("a"^(2x) - 1)/x)`
= `lim_(x -> 0)(("a"^x - 1)/x) xx lim_(x -> 0) (("a"^(2x) - 1)/(2x)) xx 2`
= `log"a"*(2log"a") ...[("As" x -> 0"," 2x -> 0 and),(lim_(x -> 0) ("a"^x - 1)/x = log "a")]`
= 2(log a)2
APPEARS IN
RELATED QUESTIONS
Evaluate the following: `lim_(x -> 0)[(9^x - 5^x)/(4^x - 1)]`
Evaluate the following: `lim_(x -> 0)[(log(2 + x) - log( 2 - x))/x]`
Evaluate the following: `lim_(x -> 0) [(2^x - 1)^2/((3^x - 1) xx log (1 + x))]`
Evaluate the following:
`lim_(x ->0)[((25)^x - 2(5)^x + 1)/x^2]`
Evaluate the following: `lim_(x -> 0)[((49)^x- 2(35)^x + (25)^x)/x^2]`
Evaluate the following Limits: `lim_(x -> 0)[(5^x - 1)/x]`
Evaluate the following Limits: `lim_(x -> 0) [("a"^(4x) - 1)/("b"^(2x) - 1)]`
Evaluate the following limit :
`lim_(x -> 0) [(6^x + 5^x + 4^x - 3^(x + 1))/sinx]`
Evaluate the following limit :
`lim_(x -> 0) [(8^sinx - 2^tanx)/("e"^(2x) - 1)]`
Evaluate the following limit :
`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
Select the correct answer from the given alternatives.
`lim_(x -> 0) [(log(5 + x) - log(5 - x))/sinx]` =
Select the correct answer from the given alternatives.
`lim_(x -> 3) [(5^(x - 3) - 4^(x - 3))/(sin(x - 3))]` =
`lim_(x -> 0) (sin^4 3x)/x^4` = ________.
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:
`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]`
\[\lim_{x\to0}\frac{\mathrm{e}^{\tan x}-\mathrm{e}^{x}}{\tan x-x}=\]
