Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x`
उत्तर
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x = lim_(x -> oo) ((x^2 - 4x + 2 + 2x - 1)/(x^2 - 4x + 2))^x`
= `lim_(x -> oo) [(x^2 - 4x - 2)/(x^2 - 4x + 2) +(2x - 1)/(x^2 - 4x + 2)]^x`
= `lim_(x -> oo) [1 + (2x - 1)/(x^2 - 4x + 2)]^x`
= `lim_(x - oo) [1 + 1/((x^2 -4x + 2)/(2x - 1))]^(((x^2 - 4x + 2)/(2x - 1) xx ((2x - 1)x)/(x^2 - 4x + 2))`
= `lim_(x -> oo) [(1 + (2x - 1)/(x^2 - 4x + 2))^((x^2 - 4x + 2)/(2x - 1))]^(((2x - 1)x)/(x^2 - 4x + 2))`
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 - 4x + 2))^x = [lim_(x -> oo) "e"]^((2x^2 - x)/(x^2 - 4x + 2))`
`lim_(x -> oo) (1 + 1/x)^x` = e
= `"e"^(lim_(x ->oo)) ((2x^2 - x)/(x^2 - 4x + 2))`
= `"e"^(lim_(x -> oo) (x^2(2 -x/x^2))/(x^2(1 - (4x)/(x^2) + 2/x^2))`
= `"e"^(lim_(x ->oo) ((2 - 1/x)/(1 -4/x + 2/x^2))`
= `"e"^(((2 - 0)/(1 - 0 + 0))`
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x` = e2
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit:
`lim_(x -> 3)[sqrt(2x + 6)/x]`
Evaluate the following limit :
`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`
Evaluate the following limit :
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
Evaluate the following limit :
If `lim_(x -> 5) [(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.
`lim_(x -> 2)(2x + 3)` = 7
Evaluate the following :
Find the limit of the function, if it exists, at x = 1
f(x) = `{(7 - 4x, "for", x < 1),(x^2 + 2, "for", x ≥ 1):}`
In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?
`lim_(x -> 3) (4 - x)`
In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?
`lim_(x -> 0) sec x`
Evaluate the following limits:
`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Evaluate the following limits:
`lim_(x -> 0) (2^x - 3^x)/x`
Evaluate the following limits:
`lim_(x -> 0) (3^x - 1)/(sqrt(x + 1) - 1)`
Evaluate the following limits:
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`
Evaluate the following limits:
`lim_(x -> pi) (1 + sinx)^(2"cosec"x)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(2) - sqrt(1 + cosx))/(sin^2x)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx`
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Choose the correct alternative:
`lim_(x -> 0) ("a"^x - "b"^x)/x` =
The value of `lim_(x→0)(sin(ℓn e^x))^2/((e^(tan^2x) - 1))` is ______.
