Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate the following limit:
`lim_(x -> 3)[sqrt(2x + 6)/x]`
Evaluate the following limit :
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.
`lim_(x -> 2)(2x + 3)` = 7
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) sin x/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.99833 | 0.99998 | 0.99999 | 0.99999 | 0.99998 | 0.99833 |
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):}`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)`
Evaluate the following limits:
`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1)`
Evaluate the following limits:
`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) ("a" > "b")`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Evaluate the following limits:
`lim_(x -> oo)(1 + 1/x)^(7x)`
Evaluate the following limits:
`lim_(x -> 0)(1 + x)^(1/(3x))`
Evaluate the following limits:
`lim_(x -> oo)(1 + "k"/x)^("m"/x)`
Evaluate the following limits:
`lim_(x -> oo) (1 + 3/x)^(x + 2)`
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/(sin 5x)`
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Choose the correct alternative:
`lim_(theta -> 0) (sinsqrt(theta))/(sqrt(sin theta)`
Choose the correct alternative:
`lim_(alpha - pi/4) (sin alpha - cos alpha)/(alpha - pi/4)` is
Choose the correct alternative:
`lim_(x -> oo) (1/"n"^2 + 2/"n"^2 + 3/"n"^2 + ... + "n"/"n"^2)` is
