Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx`
Solution
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx = lim_(x -> 0) ((sqrt(1 + sinx) sqrt(1 - sinx))(sqrt(1 + sinx) + sqrt(1 - sinx)))/(tanx(sqrt(1 - sinx) + sqrt(1 - sinx))`
= `lim_(x -> 0) ((1 + sinx) - (1 -sinx))/(sinx/cosx (sqrt(1 + sinx) + sqrt(1 - sin))`
= `lim_(x -> 0) (cosx[1 + sinx - 1 + sinx])/(sinx(sqrt(1 + sinx) + sqrt(1 - sinx))`
= `lim_(x -> 0) (cosx xx 2sinx)/(sinx(sqrt(1 + sinx) + sqrt(1 - sinx))`
= `2 lim_(x -> 0) cosx/((sqrt(1 + sinx) + sqrt(1 - sinx))`
= `2 x (cos 0)/((sqrt(1 + sin0) + sqrt(1 - sin))`
= `(2 xx 1)/((sqrt(1 + 0) + sqrt(1 - 0))`
= `2/(1 +1)`
= `2/2`
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx` = 1
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit:
`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`
Evaluate the following limit:
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
Evaluate the following limit :
If `lim_(x -> 5) [(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
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) (cos x - 1)/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.0001 | 0.01 | 0.1 |
| f(x) | 0.04995 | 0.0049999 | 0.0004999 | – 0.0004999 | – 0.004999 | – 0.04995 |
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):}`
Write a brief description of the meaning of the notation `lim_(x -> 8) f(x)` = 25
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning
Evaluate the following limits:
`lim_(x -> 5) (sqrt(x + 4) - 3)/(x - 5)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Show that `lim_("n" -> oo) (1 + 2 + 3 + ... + "n")/(3"n"^2 + 7n" + 2) = 1/6`
Evaluate the following limits:
`lim_(x -> 0) (sin^3(x/2))/x^2`
Evaluate the following limits:
`lim_(x-> 0) (1 - cos x)/x^2`
Evaluate the following limits:
`lim_(x -> 0) (1 - cos^2x)/(x sin2x)`
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"^("a"x) - "e"^("b"x))/x`
Choose the correct alternative:
`lim_(x -> oo) sinx/x`
Choose the correct alternative:
If `lim_(x -> 0) (sin "p"x)/(tan 3x)` = 4, then the value of p is
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
