Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> 0) (tan x - sin x)/x^3`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
Evaluate the following :
Find the limit of the function, if it exists, at x = 1
f(x) = `{(7 - 4x, "for", x < 1),(x^2 + 2, "for", x ≥ 1):}`
Evaluate the following :
`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`
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) (x^2 + 2)`
Sketch the graph of a function f that satisfies the given value:
f(0) is undefined
`lim_(x -> 0) f(x)` = 4
f(2) = 6
`lim_(x -> 2) f(x)` = 3
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + x) - 1)/x`
Evaluate the following limits:
`lim_(x -> 2) (2 - sqrt(x + 2))/(root(3)(2) - root(3)(4 - x))`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 - x) - 1)/x^2`
Evaluate the following limits:
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
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) (tan 2x)/x`
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 -> pi) (sin3x)/(sin2x)`
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Choose the correct alternative:
If `f(x) = x(- 1)^([1/x])`, x ≤ 0, then the value of `lim_(x -> 0) f(x)` is equal to
If `lim_(x->1)(x^5-1)/(x-1)=lim_(x->k)(x^4-k^4)/(x^3-k^3),` then k = ______.
`lim_(x→0^+)(int_0^(x^2)(sinsqrt("t"))"dt")/x^3` is equal to ______.
If `lim_(x -> 1) (x + x^2 + x^3|+ .... + x^n - n)/(x - 1)` = 820, (n ∈ N) then the value of n is equal to ______.
