Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
Advertisements
Solution
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3) = lim_(x -> oo)((2x^2 + 5 - 2)/(2x^2 + 5))^(8x^2 + 20 - 17)`
= `lim_(x -> oo) ((2x^2 - 5)/(2x^2 + 5) - 2/(2x^2 + 5))^(4(2x^2 + 5) - 17)`
= `lim_(x -> 00) (1 - 2/(2x^2 + 5))^(4(2x^2 + 5) - 17)`
Put 2x2 + 5 = y
When x → ∞
We have y = 2 × ∞ + 5 = ∞
x → ∞
⇒ y → ∞
∴ `lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3) = lim_(y -> oo) (1 - 2/y)^(4y - 17)`
= `lim_(y - oo) (1 - 2/y)^(4y) xx (1 - 2/y)^(-17)`
= `lim_(y ->oo) (1 -2/y)^(4y) xx lim_(y -> oo) (1 - 2/y)^(- 17)`
= `(lim_(y -> oo) (1 - 2/y)^y)^4 xx (1 - 2/oo)^(- 17)` ........(1)
We know `lim_(x -> oo) (1 + "k/x)^x` = ek
(1) ⇒ `lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
= `(lim_(y -> oo)(1 + ((-2))/y)^y)^4 xx (1 - 0)^(- 17)`
= `("e"^(-2))^4 xx 1`
= `"e"^(-8)`
= `1/"e"^8`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit:
`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`
Evaluate the following limit:
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
Evaluate the following limit :
`lim_(y -> 1)[(2y - 2)/(root(3)(7 + y) - 2)]`
Evaluate the following :
Given that 7x ≤ f(x) ≤ 3x2 – 6 for all x. Determine the value of `lim_(x -> 3) "f"(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`
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 -> x/2) tan x`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Evaluate the following limits:
`lim_(x -> 0) (1 - cos^2x)/(x sin2x)`
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 -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x`
Choose the correct alternative:
`lim_(x -> 0) sqrt(1 - cos 2x)/x`
Choose the correct alternative:
`lim_(x -> 0) ("a"^x - "b"^x)/x` =
Choose the correct alternative:
`lim_(x -> 3) [x]` =
Choose the correct alternative:
`lim_(alpha - pi/4) (sin alpha - cos alpha)/(alpha - pi/4)` is
Choose the correct alternative:
`lim_(x -> 0) ("e"^tanx - "e"^x)/(tan x - x)` =
`lim_(x→-1) (x^3 - 2x - 1)/(x^5 - 2x - 1)` = ______.
The value of `lim_(x rightarrow 0) (sqrt((1 + x^2)) - sqrt(1 - x^2))/x^2` is ______.
