Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> 0) (tan x - sin x)/x^3`
Advertisements
उत्तर
We know `lim_(x -> 0) sinx/x` = 1
`lim_(x -> 0) (tan x - sin x)/x^3 = lim_(x -> 0) (sinx/cosx - sin x)/x^3`
= `lim_(x -> 0) ((sinx - sinx cosx)/cosx)/x^3`
= `lim_(x -> 0) (sinx(1 - cosx))/(x^3 cosx)`
= `lim_(x -> 0) sinx/x * (2sin^2 (x/2))/(x^2) xx 1/cosx`
= `lim_(x -> 0) sinx/x xx (2sin^2 (x/2))/(2^2 xx x^2/2^2) xx 1/cosx`
= `lim_(x -> 0) sinx/x xx 1/2 (lim_(x/2 -> 0) (sin(x/2))/(x/2))^2 xx lim_(x - 0) 1/cosx`
= `1 xx 1/2 xx 1^2 xx 1/cos0`
= `1/2 xx 1/1`
`lim_(x -> 0) (tan x - sin x)/x^3 = 1/2`
APPEARS IN
संबंधित प्रश्न
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 :
`lim_(x -> 0)[x/(|x| + x^2)]`
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 -> 3) (4 - x)`
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) sin pi x`
Write a brief description of the meaning of the notation `lim_(x -> 8) f(x)` = 25
Evaluate the following limits:
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + x) - 1)/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) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Evaluate the following limits:
`lim_(x -> 0) (3^x - 1)/(sqrt(x + 1) - 1)`
Evaluate the following limits:
`lim_(x - oo){x[log(x + "a") - log(x)]}`
Evaluate the following limits:
`lim_(x -> 0) ("e"^("a"x) - "e"^("b"x))/x`
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Choose the correct alternative:
`lim_(x -> 0) ("a"^x - "b"^x)/x` =
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
The value of `lim_(x→0)(sin(ℓn e^x))^2/((e^(tan^2x) - 1))` is ______.
