Advertisements
Advertisements
प्रश्न
Evaluate the following limit :
`lim_(x -> 0)[(5x + 3)/(3 - 2x)]^(2/x)`
Advertisements
उत्तर
`lim_(x -> 0)[(5x + 3)/(3 - 2x)]^(2/x)`
= `lim_(x -> 0) [(3 + 5x)/(3 - 2x)]^(2/x)`
= `lim_(x -> 0) [(1 + (5x)/3)/(1 - (2x)/3)]^(2/x)`
= `lim_(x -> 0)((1 + (5x)/3)^(2/x))/((1 - (2x)/3)^(2/x))`
= `(lim_(x -> 0) (1 + (5x)/3)^(2/x))/(lim_(x -> 0) (1 - (2x)/3)^(2/x))`
= `(lim_(x -> 0)[(1 + (5x)/3)^(3/(5x))]^(10/3))/(lim_(x -> 0)[(1 - (2x)/3)^((-3)/(2x))]^((-4)/3)`
= `([lim_(x -> 0) (1 + (5x)/3)^(3/(5x))]^(10/3))/([lim_(x -> 0) (1 - (2x)/3)^((-3)/(2x))]^((-4)/3)`
= `("e"^(10/3))/("e"^((-4)/3)) ...[(because x -> 0 therefore (5x)/3 -> 0"," (-2x)/3 -> 0),(and lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^(14/3)`.
APPEARS IN
संबंधित प्रश्न
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) [(3^x + 3^-x - 2)/x^2]`
Evaluate the following: `lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
Evaluate the following: `lim_(x -> 0)[(log(3 - x) - log(3 + 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 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) ("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)[(4x + 1)/(1 - 4x)]^(1/x)`
Evaluate the following limit :
`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
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) ((15^x - 3^x - 5^x + 1)/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 -> pi/2) ((3^(cosx) - 1)/(pi/2 - x))` =
Select the correct answer from the given alternatives.
`lim_(x -> 3) [(5^(x - 3) - 4^(x - 3))/(sin(x - 3))]` =
Evaluate the following :
`lim_(x -> 0) [("a"^(3x) - "a"^(2x) - "a"^x + 1)/(x*tanx)]`
Evaluate the following :
`lim_(x -> 2) [(logx - log2)/(x - 2)]`
Evaluate the following :
`lim_(x -> 0) [((5^x - 1)^2)/((2^x - 1)log(1 + x))]`
If the function
f(x) = `(("e"^"kx" - 1)tan "kx")/"4x"^2, x ne 0`
= 16 , x = 0
is continuous at x = 0, then k = ?
`lim_{x→∞} ((3x + 3)^40(9x - 3)^5)/(3x + 1)^45` = ______
The value of `lim_{x→-∞} (sqrt(5x^2 + 4x + 7))/(5x + 4)` is ______
If f: R → R is defined by f(x) = [x - 2] + |x - 5| for x ∈ R, then `lim_{x→2^-} f(x)` is equal to ______
`lim_(x -> 0) (log(1 + (5x)/2))/x` is equal to ______.
`lim_(x -> 0) (sin^4 3x)/x^4` = ________.
`lim_(x -> 0) (15^x - 3^x - 5^x + 1)/(xtanx)` is equal to ______.
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 following:
`lim_(x->0)[((25)^x - 2(5)^x + 1)/x^2]`
\[\lim_{x\to0}\frac{\mathrm{e}^{\tan x}-\mathrm{e}^{x}}{\tan x-x}=\]
