Advertisements
Advertisements
प्रश्न
Evaluate the following limit :
`lim_(x -> 0) [(8^sinx - 2^tanx)/("e"^(2x) - 1)]`
उत्तर
`lim_(x -> 0) (8^sinx - 2^tanx)/("e"^(2x) - 1)`
= `lim_(x -> 0) ((8^sinx - 1)(2^tanx - 1))/("e"^(2x) - 1)`
= `lim_(x -> 0) (((8^sinx - 1) - (2^tanx - 1))/x)/(("e"^(2x) - 1)/x) ...[("Divide numerator and"),("Denominator by" x.),(because x -> 0 therefore x ≠ 0)]`
= `(lim_(x -> 0) ((8^sinx - 1)/x - (2^tanx - 1)/x))/(lim_(x -> 0) ("e"^(2x) - 1)/x)`
= `(lim_(x -> 0) ((8^sinx - 1)/sinx* sinx/x - (2^tanx - 1)/tanx * tanx/x))/(lim_(x -> 0) (e^(2x) - 1)/x)`
= `((lim_(x -> 0) (8^sinx - 1)/sinx)(lim_(x -> 0) sinx/x) - (lim_(x -> 0) (2^tanx - 1)/tanx)(lim_(x -> 0) tanx/x))/((lim_(x -> 0) ("e"^(2x) - 1)/(2x)) xx 2)`
= `(log8(1) - (log2)(1))/((1) xx 2) ...[(because x -> 0"," 2x -> 0","),(sin x -> 0"," tanx -> 0),(lim_(x -> 0) ("a"^x - 1)/x = log"a")]`
= `(log 8/2)/2`
= `(log4)/2`
= `(log(2)^2)/2`
= `(2log2)/2`
= log 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)]^(1/x)`
Evaluate the following: `lim_(x -> 0)[(log(3 - x) - log(3 + x))/x]`
Evaluate the following: `lim_(x -> 0)[(15^x - 5^x - 3^x +1)/x^2]`
Evaluate the following Limits: `lim_(x -> 0)(1 + x/5)^(1/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)[(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)[(4x + 1)/(1 - 4x)]^(1/x)`
Evaluate the following limit :
`lim_(x -> 0) [(5 + 7x)/(5 - 3x)]^(1/(3x))`
Evaluate the following limit :
`lim_(x -> 0)[(2^x - 1)^3/((3^x - 1)*sinx*log(1 + x))]`
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) ((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 -> pi/2) ((3^(cosx) - 1)/(pi/2 - x))` =
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 -> 1) [("ab"^x - "a"^x"b")/(x^2 - 1)]`
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 = ?
lf the function f(x) satisfies `lim_{x→1}(2f(x) - 5)/(2(x^2 - 1)) = e`, then `lim_{x→1}f(x)` is ______
`lim_(x -> 0) (sin^4 3x)/x^4` = ________.
The value of `lim_{x→0} (1 + sinx - cosx + log_e(1 - x))/x^3` is ______
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->1)[(x^3-1)/(x^2+5x-6)]`
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]`
