Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> 0) (sqrt(2) - sqrt(1 + cosx))/(sin^2x)`
Advertisements
Solution
`lim_(x -> 0) ((sqrt(2) - sqrt(1 + cosx))/(sin^2x))`
= `lim_(x -> 0) ((sqrt(2) - sqrt(1 + cosx))(sqrt(2) + sqrt(1 + cosx)))/(sin^2x (sqrt(2) + sqrt(1 + cosx))`
= `lim_(x -> 0) (2 - (1 + cosx))/((1 - cos^2x)(sqrt(2) + sqrt(1 + cosx))`
= `lim_(x -> 0) (1 - cosx)/((1 +cosx)(1 - cosx)(sqrt(2) + sqrt(1 + cosx))`
= `lim_(x -> 0) 1/((1 + cosx)(sqrt(2) + sqrt(1 + cosx))`
= `1/((1 + cos0)(sqrt(2) + sqrt(1 + 1))`
= `1/(2(sqrt(2) + sqrt(2))`
=`1/(2 xx 2sqrt(2))`
`lim_(x -> 0) (sqrt(2) - sqrt(1 + cosx))/(sin^2x) = 1/(4sqrt(2))`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit:
`lim_(z -> -3) [sqrt("z" + 6)/"z"]`
Evaluate the following limit:
If `lim_(x -> 1)[(x^4 - 1)/(x - 1)]` = `lim_(x -> "a")[(x^3 - "a"^3)/(x - "a")]`, find all possible values of a
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_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`
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
Evaluate the following :
`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`
Sketch the graph of a function f that satisfies the given value:
f(– 2) = 0
f(2) = 0
`lim_(x -> 2) f(x)` = 0
`lim_(x -> 2) f(x)` does not exist.
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 -> 2) (2 - sqrt(x + 2))/(root(3)(2) - root(3)(4 - x))`
Evaluate the following limits:
`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) ("a" > "b")`
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 -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`
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 -> oo)(1 + "k"/x)^("m"/x)`
Evaluate the following limits:
`lim_(x -> 0) (sin^3(x/2))/x^2`
Evaluate the following limits:
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x`
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)` =
`lim_(x -> 5) |x - 5|/(x - 5)` = ______.
