Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`
Advertisements
उत्तर
We know `lim_(x -> 0) ("e"^x - 1)/x` = 1
`lim_(x -> 0) ("a"^x - 1)/x` = log a
`lim_(x -> 0) (1 - cosx)/x` = 0
`lim_(x -> oo) [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)] = lim_(x -> oo) [(3^(1/x) + 1 - cos(1/x) - "e"^(1/x))/(1/x)]`
= `lim_(x -> oo) [(3^(1/x) - 1 + 1 - "e"^(1/x))/(1/x) + (1 - cos(1/x))/(1/x)]`
= `lim_(x -> oo) [((3^(1/x) - 1) - ("e"^(1/x) - 1))/(1/x) + (1 - cos(1/x))/(1/x)]`
= `lim_(x > 0)[(3^(1/x) - 1)/(1/x) - ("e"^(1/x) - 1)/(1/x) + (1 -cos(1/x))/(1/x)]`
Put y = `1/x`
When x = `oo`
⇒ y = `1/oo` = 0
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)] = lim_(y - 0) [(3y - 1)/y - ("e"^y - 1)/y + (1 - cosy)/y]`
= `(lim_(y -> 0) (3^y - 1)/y) -(lim_(y -> 0) ("e"^y - 1)/y) + (lim_(y -> 0) (1 - cosy)/y)`
= `log 3 - 1 + 0`
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)] = (log 3) - 1`
APPEARS IN
संबंधित प्रश्न
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 -> 2) f(x)` where `f(x) = {{:(4 - x",", x ≠ 2),(0",", x = 2):}`
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):}`
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):}`
Write a brief description of the meaning of the notation `lim_(x -> 8) f(x)` = 25
Evaluate the following limits:
`lim_(x - 0) (sqrt(1 + x^2) - 1)/x`
Find the left and right limits of f(x) = tan x at x = `pi/2`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Evaluate the following limits:
`lim_(x -> oo) (1 + 3/x)^(x + 2)`
Evaluate the following limits:
`lim_(x -> 0) (sin^3(x/2))/x^2`
Evaluate the following limits:
`lim_(x -> 0) (sinalphax)/(sinbetax)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(x^2 + "a"^2) - "a")/(sqrt(x^2 + "b"^2) - "b")`
Evaluate the following limits:
`lim_(x-> 0) (1 - cos x)/x^2`
Choose the correct alternative:
`lim_(theta -> 0) (sinsqrt(theta))/(sqrt(sin theta)`
Choose the correct alternative:
`lim_(x -> 0) (x"e"^x - sin x)/x` is
Choose the correct alternative:
If `lim_(x -> 0) (sin "p"x)/(tan 3x)` = 4, then the value of p is
If `lim_(x->1)(x^5-1)/(x-1)=lim_(x->k)(x^4-k^4)/(x^3-k^3),` then k = ______.
If `lim_(x -> 1) (x + x^2 + x^3|+ .... + x^n - n)/(x - 1)` = 820, (n ∈ N) then the value of n is equal to ______.
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
