Advertisements
Advertisements
Question
`lim_(x -> pi/4) (tan^3x - tan x)/(cos(x + pi/4))`
Advertisements
Solution
Given, `lim_(x -> pi/4) (tan^3x - tan x)/(cos(x + pi/4))`
= `lim_(x -> pi/4) (tanx(tan^2x - 1))/(cos(x + pi/4))`
= `lim_(x -> pi/4) (tanx (tan^2x - 1))/(cos(x + pi/4))`
= `lim_(x -> pi/4) tan x * lim_(x -> pi/4) [(-(1 - tan^2x))/(cos(x + pi/4))]`
= `-1 xx lim_(x -> pi/4) ((1 - tanx)(1 + tanx))/(cos(x + pi/4))`
= `lim_(x -> pi/4) - (1 + tan x) * lim_(x -> pi/4) ((1 - tanx)/(cos(x + pi/4)))`
= `-(1 + 1) * lim_(x -> pi/4) ((cosx - sin x))/(cosx * cos(x + pi/4))`
= `-2 xx lim_(x -> pi/4) (sqrt(2) (1/sqrt(2) cos x - 1/sqrt(2) sinx))/(cos x * cos (x + pi/4))`
= `-2sqrt(2) lim_(x -> pi/4) ([cos pi/4 * cos x - sin pi/4 sin x])/(cosx * cos(x + pi/4))`
= `lim_(x -> pi/4) (-2sqrt(2) * cos(x + pi/4))/(cosx * cos(x + pi/4))`
= `(-2sqrt(2))/(cos pi/4)` ....(Taking limit)
= `(-2sqrt(2))/(1/sqrt(2))`
= `-2 xx 2`
= – 4.
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit.
`lim_(x ->0) cos x/(pi - x)`
Evaluate the following limit.
`lim_(x -> 0) (sin ax + bx)/(ax + sin bx) a, b, a+ b != 0`
Evaluate the following limit :
`lim_(x -> pi) [(sqrt(1 - cosx) - sqrt(2))/(sin^2 x)]`
Select the correct answer from the given alternatives.
`lim_(x -> pi/2) [(3cos x + cos 3x)/(2x - pi)^3]` =
Evaluate the following :
`lim_(x -> 0)[(secx^2 - 1)/x^4]`
Evaluate `lim_(x -> 2) 1/(x - 2) - (2(2x - 3))/(x^3 - 3x^2 + 2x)`
Evaluate `lim_(x -> pi/6) (2sin^2x + sin x - 1)/(2sin^2 x - 3sin x + 1)`
Find the derivative of f(x) = `sqrt(sinx)`, by first principle.
`lim_(x -> pi/2) (1 - sin x)/cosx` is equal to ______.
Evaluate: `lim_(x -> 3) (x^2 - 9)/(x - 3)`
Evaluate: `lim_(x -> 1) (x^4 - sqrt(x))/(sqrt(x) - 1)`
Evaluate: `lim_(x -> 3) (x^3 + 27)/(x^5 + 243)`
Evaluate: `lim_(x -> 1/2) (8x - 3)/(2x - 1) - (4x^2 + 1)/(4x^2 - 1)`
Evaluate: `lim_(x -> 0) (sin 3x)/(sin 7x)`
Evaluate: `lim_(x -> 0) (1 - cos mx)/(1 - cos nx)`
Evaluate: `lim_(x -> pi/3) (sqrt(1 - cos 6x))/(sqrt(2)(pi/3 - x))`
Evaluate: `lim_(x -> 0) (sin 2x + 3x)/(2x + tan 3x)`
Evaluate: `lim_(x -> a) (sin x - sin a)/(sqrt(x) - sqrt(a))`
cos (x2 + 1)
`x^(2/3)`
`lim_(y -> 0) ((x + y) sec(x + y) - x sec x)/y`
`lim_(x -> 0) ((sin(alpha + beta) x + sin(alpha - beta)x + sin 2alpha x))/(cos 2betax - cos 2alphax) * x`
`lim_(x -> 0) ((1 + x)^n - 1)/x` is equal to ______.
`lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(2x^2 + x - 3)` is ______.
If `f(x) = {{:(sin[x]/[x]",", [x] ≠ 0),(0",", [x] = 0):}`, where [.] denotes the greatest integer function, then `lim_(x -> 0) f(x)` is equal to ______.
If `f(x) = {{:(x^2 - 1",", 0 < x < 2),(2x + 3",", 2 ≤ x < 3):}`, the quadratic equation whose roots are `lim_(x -> 2^-) f(x)` and `lim_(x -> 2^+) f(x)` is ______.
`lim_(x -> 0) (tan 2x - x)/(3x - sin x)` is equal to ______.
`lim_(x -> 3^+) x/([x])` = ______.
The value of `lim_(x → ∞) ((x^2 - 1)sin^2(πx))/(x^4 - 2x^3 + 2x - 1)` is equal to ______.
If `lim_(x→∞) 1/(x + 1) tan((πx + 1)/(2x + 2)) = a/(π - b)(a, b ∈ N)`; then the value of a + b is ______.
