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)[(9^x - 5^x)/(4^x - 1)]`
Evaluate the following: `lim_(x -> 0) [("a"^(3x) - "b"^(2x))/(log 1 + 4x)]`
Evaluate the following: `lim_(x -> 0)[(15^x - 5^x - 3^x +1)/x^2]`
Evaluate the following Limits: `lim_(x -> 0)[(x(6^x - 3^x))/((2^x - 1)*log(1 + x))]`
Evaluate the following limit :
`lim_(x -> 0) [(6^x + 5^x + 4^x - 3^(x + 1))/sinx]`
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) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
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 -> pi/2) ((3^(cosx) - 1)/(pi/2 - x))` =
Select the correct answer from the given alternatives.
`lim_(x -> 0) [(x*log(1 + 3x))/("e"^(3x) - 1)^2]` =
Evaluate the following :
`lim_(x -> 0) [("a"^(3x) - "a"^(2x) - "a"^x + 1)/(x*tanx)]`
The value of `lim_{x→0}{(a^x + b^x + c^x + d^x)/4}^{1/x}` is ______
`lim_{x→∞} ((3x + 3)^40(9x - 3)^5)/(3x + 1)^45` = ______
The value of `lim_{x→-∞} (sqrt(5x^2 + 4x + 7))/(5x + 4)` is ______
`lim_(x -> 0) (sin^4 3x)/x^4` = ________.
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]`
