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_(z -> -3) [sqrt("z" + 6)/"z"]`
Evaluate the following limit :
`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`
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 |
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 -> 1) f(x)` where `f(x) = {{:(x^2 + 2",", x ≠ 1),(1",", 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 -> 1) sin pi 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`
Sketch the graph of f, then identify the values of x0 for which `lim_(x -> x_0)` f(x) exists.
f(x) = `{{:(sin x",", x < 0),(1 - cos x",", 0 ≤ x ≤ pi),(cos x",", x > pi):}`
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning
Evaluate : `lim_(x -> 3) (x^2 - 9)/(x - 3)` if it exists by finding `f(3^-)` and `f(3^+)`
Evaluate the following limits:
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1)`, m and n are integers
Evaluate the following limits:
`lim_(x -> oo) (x^3 + x)/(x^4 - 3x^2 + 1)`
Show that `lim_("n" -> oo) 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/("n"("n" + 1))` = 1
Evaluate the following limits:
`lim_(alpha -> 0) (sin(alpha^"n"))/(sin alpha)^"m"`
Choose the correct alternative:
`lim_(x -> oo) ((x^2 + 5x + 3)/(x^2 + x + 3))^x` is
Choose the correct alternative:
`lim_(x - oo) sqrt(x^2 - 1)/(2x + 1)` =
Choose the correct alternative:
`lim_(x -> 0) ("a"^x - "b"^x)/x` =
If `lim_(x->1)(x^5-1)/(x-1)=lim_(x->k)(x^4-k^4)/(x^3-k^3),` then k = ______.
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
If f(x) is differentiable at x = 1 and `lim_(h → 0) 1/h f(1 + h) = 5`, then f' (1) is equal to ______.
