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 -> 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 -> 0)[((1 - x)^8 - 1)/((1 - x)^2 - 1)]`
Evaluate the following limit :
`lim_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`
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]`
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (sqrt(x + 3) - sqrt(3))/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.2911 | 0.2891 | 0.2886 | 0.2886 | 0.2885 | 0.28631 |
Sketch the graph of f, then identify the values of x0 for which `lim_(x -> x_0)` f(x) exists.
f(x) = `{{:(x^2",", x ≤ 2),(8 - 2x",", 2 < x < 4),(4",", x ≥ 4):}`
Evaluate the following limits:
`lim_(x - 0) (sqrt(1 + x^2) - 1)/x`
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
Evaluate the following limits:
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
Evaluate the following limits:
`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`
Evaluate the following limits:
`lim_(x-> 0) (1 - cos x)/x^2`
Evaluate the following limits:
`lim_(x -> 0) ("e"^x - "e"^(-x))/sinx`
Choose the correct alternative:
`lim_(theta -> 0) (sinsqrt(theta))/(sqrt(sin theta)`
Choose the correct alternative:
`lim_(x -> oo) (1/"n"^2 + 2/"n"^2 + 3/"n"^2 + ... + "n"/"n"^2)` is
`lim_(x -> 5) |x - 5|/(x - 5)` = ______.
`lim_(x -> 0) (sin 4x + sin 2x)/(sin5x - sin3x)` = ______.
`lim_(x→0^+)(int_0^(x^2)(sinsqrt("t"))"dt")/x^3` is equal to ______.
