Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx`
Advertisements
उत्तर
`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
संबंधित प्रश्न
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_(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 -> -3) (3x + 2)` = – 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 |
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):}`
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 -> x/2) tan x`
Show that `lim_("n" -> oo) 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/("n"("n" + 1))` = 1
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/(sin 5x)`
Evaluate the following limits:
`lim_(x -> 0) (2 "arc"sinx)/(3x)`
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/x`
Evaluate the following limits:
`lim_(x - oo){x[log(x + "a") - log(x)]}`
Evaluate the following limits:
`lim_(x -> 0) ("e"^x - "e"^(-x))/sinx`
Evaluate the following limits:
`lim_(x -> 0) (tan x - sin x)/x^3`
Choose the correct alternative:
`lim_(x -> oo) ((x^2 + 5x + 3)/(x^2 + x + 3))^x` is
Choose the correct alternative:
`lim_(x -> 0) ("e"^tanx - "e"^x)/(tan x - x)` =
`lim_(x -> 0) (sin 4x + sin 2x)/(sin5x - sin3x)` = ______.
