Advertisements
Advertisements
Question
Evaluate the following :
`lim_(x -> 1) [("ab"^x - "a"^x"b")/(x^2 - 1)]`
Solution
`lim_(x -> 1) ("ab"^x - "a"^x"b")/(x^2 - 1)`
= `lim_(x -> 1) ("ab"("b"^(x - 1) - "a"^(x - 1)))/(x^2 - 1^2)`
= `lim_(x -> 1) ("ab"("b"^(x - 1) - "a"^(x - 1)))/((x - 1)(x + 1))`
Put x = 1 + h, ∴ x – 1 = h
As x → 1, h → 0
∴ `lim_(x -> 1) ("ab"^x - "a"^x"b")/(x^2 - 1)`
= `lim_("h" -> 0) ("ab"("b"^"h" - "a"^"h"))/("h"(1 + "h" + 1))`
= `"ab" lim_("h" -> 0) ("b"^"h" - 1 + 1 - "a"^"h")/("h"(2 + "h"))`
= `"ab" lim_("h" -> 0) (("b"^"h" - 1) - ("a"^"h" - 1))/("h"(2 + "h"))`
= `"ab" lim_("h" -> 0) 1/(2 + "h") (("b"^"h" - 1)/"h" - ("a"^"h" - 1)/"h")`
= `"ab"* 1/(lim_("h" -> 0)(2 + "h")) (lim_("h" -> 0) ("b"^"h" - 1)/"h" - lim_("h" -> 0) ("a"^"h" - 1)/"h")`
= `"ab"* 1/(2 + 0) * (log"b" - log"a") ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `"ab"/2 log ("b"/"a")`
APPEARS IN
RELATED QUESTIONS
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 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)[(x(6^x - 3^x))/((2^x - 1)*log(1 + x))]`
Evaluate the following Limits: `lim_(x -> 0)[("a"^(3x) - "a"^(2x) - "a"^x + 1)/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) [(8^sinx - 2^tanx)/("e"^(2x) - 1)]`
Evaluate the following limit :
`lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
Evaluate the following limit :
`lim_(x -> 0) [(log(3 - x) - log(3 + x))/x]`
Evaluate the following limit :
`lim_(x -> 0)[(15^x - 5^x - 3^x + 1)/(x*sinx)]`
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) ((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)[(3^(sinx) - 1)^3/((3^x - 1).tan x.log(1 + x))]` =
Evaluate the following :
`lim_(x -> 0) [("a"^(3x) - "a"^(2x) - "a"^x + 1)/(x*tanx)]`
Evaluate the following :
`lim_(x -> 0) [((5^x - 1)^2)/((2^x - 1)log(1 + x))]`
lf the function f(x) satisfies `lim_{x→1}(2f(x) - 5)/(2(x^2 - 1)) = e`, then `lim_{x→1}f(x)` is ______
The value of `lim_{x→2} (e^{3x - 6} - 1)/(sin(2 - x))` is ______
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 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]`
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]`
