Advertisements
Advertisements
Question
Evaluate the following :
`lim_(x -> 0) [(sqrt(1 - cosx))/x]`
Advertisements
Solution
Let f(x) = `(sqrt(1 - cosx))/x`
= `(sqrt(1 - cosx))/x xx (sqrt(1 + cosx))/(sqrt(1 + cos x))`
= `(sqrt(1 - cos^2x))/(xsqrt(1 + cosx))`
= `(sqrt(sin^2x))/(xsqrt(1 + cos x))`
= `|sinx|/(xsqrt(1 + cos x)`
Now, |sin x| = `{(sinx, "if" x > 0),(-sinx, "if" x < 0):}`
∴ `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) (sqrt(1 - cosx))/x`
= `lim_(x -> 0) sinx/(xsqrt(1 + cosx))`
= `(lim_(x -> 0) (sinx/x))/(lim_(x -> 0) sqrt(1 + cos x))`
= `1/(sqrt(1 + cos 0))`
= `1/sqrt(2)` ...(1)
∴ `lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (sqrt(1 - cosx))/x`
= `lim_(x -> 0) (- sinx)/(xsqrt(1 + cosx))`
= `-lim_(x -> 0) sinx/(xsqrt(1 + cos x)`
=`-lim_(x-> 0) ((sinx/x))/sqrt(1 + cos x)`
= `- (lim_(x -> 0) (sinx/x))/(lim_(x -> 0) (sqrt(1 + cos x))`
= `- 1/sqrt(1 + 1)`
= `-1/sqrt(2)` ...(2)
From (1) and (2),
`lim_(x -> 0^+) "f"(x) ≠ lim_(x -> 0^-) "f"(x)`
∴ `lim_(x -> 0) "f"(x) "i.e.", lim_(x -> 0) (sqrt(1 - cos x))/x` does not exist.
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit:
`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`
Evaluate the following limit:
`lim_(x -> 3)[sqrt(2x + 6)/x]`
Evaluate the following limit:
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
Evaluate the following limit :
`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
Evaluate the following limit :
`lim_(x -> 1) [(x + x^3 + x^5 + ... + x^(2"n" - 1) - "n")/(x - 1)]`
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) (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 |
If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?
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_(x -> 5) (sqrt(x + 4) - 3)/(x - 5)`
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 -> 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 ->oo) (x^3/(2x^2 - 1) - x^2/(2x + 1))`
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)(1 + x)^(1/(3x))`
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
Evaluate the following limits:
`lim_(x -> 0) (sin^3(x/2))/x^2`
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`
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/x`
Evaluate the following limits:
`lim_(x -> 0) (2^x - 3^x)/x`
Evaluate the following limits:
`lim_(x -> 0) (3^x - 1)/(sqrt(x + 1) - 1)`
Evaluate the following limits:
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`
Evaluate the following limits:
`lim_(x -> 0) ("e"^x - "e"^(-x))/sinx`
Evaluate the following limits:
`lim_(x -> 0) ("e"^("a"x) - "e"^("b"x))/x`
Choose the correct alternative:
`lim_(x -> oo) sinx/x`
Choose the correct alternative:
`lim_(x - pi/2) (2x - pi)/cos x`
Choose the correct alternative:
If `lim_(x -> 0) (sin "p"x)/(tan 3x)` = 4, then the value of p is
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 ______.
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
