Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
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 -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`
Evaluate the following limit :
`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`
Evaluate the following :
Given that 7x ≤ f(x) ≤ 3x2 – 6 for all x. Determine the value of `lim_(x -> 3) "f"(x)`
Evaluate the following :
`lim_(x -> 0) [(sqrt(1 - cosx))/x]`
Verify the existence of `lim_(x -> 1) f(x)`, where `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`
Evaluate the following limits:
`lim_("h" -> 0) (sqrt(x + "h") - sqrt(x))/"h", x > 0`
Find the left and right limits of f(x) = tan x at x = `pi/2`
Evaluate the following limits:
`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`
Show that `lim_("n" -> oo) (1^2 + 2^2 + ... + (3"n")^2)/((1 + 2 + ... + 5"n")(2"n" + 3)) = 9/25`
Show that `lim_("n" -> oo) 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/("n"("n" + 1))` = 1
A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tank at a rate of 25 litres per minute. The concentration of salt water after t minutes (in grams per litre) is C(t) = `(30"t")/(200 + "t")`. What happens to the concentration as t → ∞?
Evaluate the following limits:
`lim_(x -> 0) ("e"^("a"x) - "e"^("b"x))/x`
Choose the correct alternative:
`lim_(theta -> 0) (sinsqrt(theta))/(sqrt(sin theta)`
Choose the correct alternative:
`lim_(x -> 0) ("a"^x - "b"^x)/x` =
Choose the correct alternative:
If `f(x) = x(- 1)^([1/x])`, x ≤ 0, then the value of `lim_(x -> 0) f(x)` is equal to
Choose the correct alternative:
`lim_(x -> 3) [x]` =
`lim_(x -> 0) ((2 + x)^5 - 2)/((2 + x)^3 - 2)` = ______.
