Advertisements
Advertisements
प्रश्न
Evaluate the following limit :
`lim_(x -> 0) [((49)^x - 2(35)^x + (25)^x)/(sinx* log(1 + 2x))]`
Advertisements
उत्तर
`lim_(x -> 0) [((49)^x - 2(35)^x + (25)^x)/(sinx* log(1 + 2x))]`
= `lim_(x -> 0) ((7^x)^2 - 2(7^x) (5^x) + (5^x)^2)/(sinx log (1 + 2x)) ...[(35^x = (5*7)^x = 5^x * 7^x),(49^x = (7^2)x = (7^x)^2)]`
= `lim_(x -> 0) [7^x - 5^x]^2/(sinx log (1 + 2x)`
= `lim_(x -> 0) [(7^x - 1) - (5^x- 1)]^2/(sin x log (1 + 2x))`
= `lim_(x -> 0) [(7^x - 1)/x - (5^x - 1)/x]^2/([sinx/x][(log(1 + 2x))/x])` ...[∵ x → 0, ∴ x ≠ 0]
= `([lim_(x -> 0) ((7^x - 1)/x - (5^x - 1)/x)]^2)/([lim_(x -> 0) sinx/x] xx 2 [lim_(x -> 0) (log(1 + 2x))/(2x)]`
= `[log 7 - log 5]^2/([1] xx 2[1]) ...[because x -> 0 "," 2x -> 0 "and" lim_(x -> 0) (log(1 + x))/x = 1 "and" lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `1/2[log(7/5)]^2`
APPEARS IN
संबंधित प्रश्न
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)[(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)[(15^x - 5^x - 3^x +1)/x^2]`
Evaluate the following:
`lim_(x ->0)[((25)^x - 2(5)^x + 1)/x^2]`
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)[("a"^(3x) - "a"^(2x) - "a"^x + 1)/x^2]`
Evaluate the following Limits: `lim_(x -> 0) [("a"^(4x) - 1)/("b"^(2x) - 1)]`
Evaluate the following limit :
`lim_(x -> 0)[("a"^x + "b"^x + "c"^x - 3)/sinx]`
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) [(5 + 7x)/(5 - 3x)]^(1/(3x))`
Evaluate the following limit :
`lim_(x -> 0)[(15^x - 5^x - 3^x + 1)/(x*sinx)]`
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) ((3 + 5x)/(3 - 4x))^(1/x)` =
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→0)[(3^(sinx) - 1)^3/((3^x - 1).tan x.log(1 + x))]` =
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))]`
The value of `lim_{x→0}{(a^x + b^x + c^x + d^x)/4}^{1/x}` is ______
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 ______.
The value of `lim_{x→2} (e^{3x - 6} - 1)/(sin(2 - x))` is ______
`lim_(x -> 0) (15^x - 3^x - 5^x + 1)/(xtanx)` is equal to ______.
Evaluate the following limit :
`lim_(x->0)[(sqrt(6+x+x^2)-sqrt6)/x]`
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]`
Evaluate the following:
`lim_(x->0)[((25)^x -2(5)^x +1)/(x^2)]`
