Advertisements
Advertisements
Question
Evaluate the following: `lim_(x -> 0)[(log(2 + x) - log( 2 - x))/x]`
Advertisements
Solution
`lim_(x -> 0)[(log(2 + x) - log( 2 - x))/x]`
= `lim_(x -> 0)(log[2(1 + x/2)] - log[2(1 - x/2)])/x`
= `lim_(x -> 0) (log2 + log(1 + x/2) - [log2 + log(1 - x/2)])/x`
= `lim_(x -> 0) (log(1 + x/2) - log(1 - x/2))/x`
= `lim_(x -> 0) [(log(1 + x/2))/x - (log(1 - x/2))/x]`
= `lim_(x -> 0) [(log(1 + x/2))/(2(x/2)) - (log(1 - x/2))/((-2)(-x/2))]`
= `1/2 lim_(x -> 0) (log(1 + x/2))/(x/2) + 1/2 lim_(x -> 0) (log(1 - x/2))/(-x/2)`
= `1/2(1) + 1/2(1) ....[(because x -> 0"," x/2 -> 0 and),(lim_(x -> 0) (log(1 + x))/x = 1)]`
= 1
APPEARS IN
RELATED QUESTIONS
Evaluate the following: `lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
Evaluate the following: `lim_(x -> 0) [(2^x - 1)^2/((3^x - 1) xx log (1 + x))]`
Evaluate the following Limits: `lim_(x -> 0)((1 - x)^5 - 1)/((1 - x)^3 - 1)`
Evaluate the following limit :
`lim_(x -> 0)[("a"^x + "b"^x + "c"^x - 3)/sinx]`
Evaluate the following limit :
`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
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) [((25)^x - 2(5)^x + 1)/(x*sinx)]`
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→0)[(3^(sinx) - 1)^3/((3^x - 1).tan x.log(1 + 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 -> 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 = ?
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 ______
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) (log(1 + (5x)/2))/x` is equal to ______.
The value of `lim_{x→2} (e^{3x - 6} - 1)/(sin(2 - x))` is ______
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]`
